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Autonomic nervous system
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Autonomic nervous system
Autonomic nervous system innervation.
Details
Identifiers
Latin
Autonomici systematis nervosi
TA
A14.3.00.001
FMA
9905
Anatomical terminology
[edit on Wikidata]
The autonomic nervous system (ANS), formerly the vegetative nervous system, is a division of the peripheral nervous system that supplies smooth muscle and glands, and thus influences the function of internal organs.[1] The autonomic nervous system is a control system that acts largely unconsciously and regulates bodily functions such as the heart rate, digestion, respiratory rate, pupillary response, urination, and sexual arousal.[2] This system is the primary mechanism in control of the fight-or-flight response.
Within the brain, the autonomic nervous system is regulated by the hypothalamus. Autonomic functions include control of respiration, cardiac regulation (the cardiac control center), vasomotor activity (the vasomotor center), and certain reflex actions such as coughing, sneezing, swallowing and vomiting. Those are then subdivided into other areas and are also linked to ANS subsystems and nervous systems external to the brain. The hypothalamus, just above the brain stem, acts as an integrator for autonomic functions, receiving ANS regulatory input from the limbic system to do so.[3]
The autonomic nervous system has three branches: the sympathetic nervous system, the parasympathetic nervous system and the enteric nervous system.[4][5][6][7] Some textbooks do not include the enteric nervous system as part of this system.[8] The sympathetic nervous system is often considered the "fight or flight" system, while the parasympathetic nervous system is often considered the "rest and digest" or "feed and breed" system. In many cases, both of these systems have "opposite" actions where one system activates a physiological response and the other inhibits it. An older simplification of the sympathetic and parasympathetic nervous systems as "excitory" and "inhibitory" was overturned due to the many exceptions found. A more modern characterization is that the sympathetic nervous system is a "quick response mobilizing system" and the parasympathetic is a "more slowly activated dampening system", but even this has exceptions, such as in sexual arousal and orgasm, wherein both play a role.[3]
There are inhibitory and excitatory synapses between neurons. Relatively recently, a third subsystem of neurons that have been named non-noradrenergic, non-cholinergic transmitters (because they use nitric oxide as a neurotransmitter) have been described and found to be integral in autonomic function, in particular in the gut and the lungs.[9]
Although the ANS is also known as the visceral nervous system, the ANS is only connected with the motor side.[10] Most autonomous functions are involuntary but they can often work in conjunction with the somatic nervous system which provides voluntary control.
Contents
[hide]
1 Structure
1.1 Sympathetic division
1.2 Parasympathetic division
1.3 Sensory neurons
1.4 Innervation
1.5 Motor neurons
2 Function
2.1 Sympathetic nervous system
2.2 Parasympathetic nervous system
2.3 Enteric nervous system
2.4 Neurotransmitters
3 Caffeine effects
4 See also
5 References
6 External links
Structure[edit]
Autonomic nervous system, showing splanchnic nerves in middle, and the vagus nerve as "X" in blue. The heart and organs below in list to right are regarded as viscera.
The autonomic nervous system is divided into the sympathetic nervous system and parasympathetic nervous system. The sympathetic division emerges from the spinal cord in the thoracic and lumbar areas, terminating around L2-3. The parasympathetic division has craniosacral “outflow”, meaning that the neurons begin at the cranial nerves (specifically the oculomotor nerve, facial nerve, glossopharyngeal nerve and vagus nerve) and sacral (S2-S4) spinal cord.
The autonomic nervous system is unique in that it requires a sequential two-neuron efferent pathway; the preganglionic neuron must first synapse onto a postganglionic neuron before innervating the target organ. The preganglionic, or first, neuron will begin at the “outflow” and will synapse at the postganglionic, or second, neuron’s cell body. The postganglionic neuron will then synapse at the target organ.
Sympathetic division[edit]
Main article: Sympathetic nervous system
The sympathetic nervous system consists of cells with bodies in the lateral grey column from T1 to L2/3. These cell bodies are "GVE" (general visceral efferent) neurons and are the preganglionic neurons. There are several locations upon which preganglionic neurons can synapse for their postganglionic neurons:
Paravertebral ganglia (3) of the sympathetic chain (these run on either side of the vertebral bodies)
cervical ganglia (3)
thoracic ganglia (12) and rostral lumbar ganglia (2 or 3)
caudal lumbar ganglia and sacral ganglia
Prevertebral ganglia (celiac ganglion, aorticorenal ganglion, superior mesenteric ganglion, inferior mesenteric ganglion)
Chromaffin cells of the adrenal medulla (this is the one exception to the two-neuron pathway rule: the synapse is directly efferent onto the target cell bodies)
These ganglia provide the postganglionic neurons from which innervation of target organs follows. Examples of splanchnic (visceral) nerves are:
Cervical cardiac nerves & thoracic visceral nerves, which synapse in the sympathetic chain
Thoracic splanchnic nerves (greater, lesser, least), which synapse in the prevertebral ganglia
Lumbar splanchnic nerves, which synapse in the prevertebral ganglia
Sacral splanchnic nerves, which synapse in the inferior hypogastric plexus
These all contain afferent (sensory) nerves as well, known as GVA (general visceral afferent) neurons.
Parasympathetic division[edit]
Main article: Parasympathetic nervous system
The parasympathetic nervous system consists of cells with bodies in one of two locations: the brainstem (Cranial Nerves III, VII, IX, X) or the sacral spinal cord (S2, S3, S4). These are the preganglionic neurons, which synapse with postganglionic neurons in these locations:
Parasympathetic ganglia of the head: Ciliary (Cranial nerve III), Submandibular (Cranial nerve VII), Pterygopalatine (Cranial nerve VII), and Otic (Cranial nerve IX)
In or near the wall of an organ innervated by the Vagus (Cranial nerve X) or Sacral nerves (S2, S3, S4)
These ganglia provide the postganglionic neurons from which innervations of target organs follows. Examples are:
The postganglionic parasympathetic splanchnic (visceral) nerves
The vagus nerve, which passes through the thorax and abdominal regions innervating, among other organs, the heart, lungs, liver and stomach
Sensory neurons[edit]
Main article: Sensory neuron
The sensory arm is composed of primary visceral sensory neurons found in the peripheral nervous system (PNS), in cranial sensory ganglia: the geniculate, petrosal and nodose ganglia, appended respectively to cranial nerves VII, IX and X. These sensory neurons monitor the levels of carbon dioxide, oxygen and sugar in the blood, arterial pressure and the chemical composition of the stomach and gut content. They also convey the sense of taste and smell, which, unlike most functions of the ANS, is a conscious perception. Blood oxygen and carbon dioxide are in fact directly sensed by the carotid body, a small collection of chemosensors at the bifurcation of the carotid artery, innervated by the petrosal (IXth) ganglion. Primary sensory neurons project (synapse) onto “second order” visceral sensory neurons located in the medulla oblongata, forming the nucleus of the solitary tract (nTS), that integrates all visceral information. The nTS also receives input from a nearby chemosensory center, the area postrema, that detects toxins in the blood and the cerebrospinal fluid and is essential for chemically induced vomiting or conditional taste aversion (the memory that ensures that an animal that has been poisoned by a food never touches it again). All this visceral sensory information constantly and unconsciously modulates the activity of the motor neurons of the ANS.
Innervation[edit]
Autonomic nerves travel to organs throughout the body. Most organs receive parasympathetic supply by the vagus nerve and sympathetic supply by splanchnic nerves. The sensory part of the latter reaches the spinal column at certain spinal segments. Pain in any internal organ is perceived as referred pain, more specifically as pain from the dermatome corresponding to the spinal segment.[11]

Autonomic nervous supply to organs in the human body edit
Organ
Nerves[12]
Spinal column origin[12]
stomach
PS: anterior and posterior vagal trunks
S: greater splanchnic nerves
T6, T7, T8, T9, sometimes T10
duodenum
PS: vagus nerves
S: greater splanchnic nerves
T5, T6, T7, T8, T9, sometimes T10
jejunum and ileum
PS: posterior vagal trunks
S: greater splanchnic nerves
T5, T6, T7, T8, T9
spleen
S: greater splanchnic nerves
T6, T7, T8
gallbladder and liver
PS: vagus nerve
S: celiac plexus
right phrenic nerve
T6, T7, T8, T9
colon
PS: vagus nerves and pelvic splanchnic nerves
S: lesser and least splanchnic nerves
T10, T11, T12 (proximal colon)
L1, L2, L3, (distal colon)
pancreatic head
PS: vagus nerves
S: thoracic splanchnic nerves
T8, T9
vermiform appendix
nerves to superior mesenteric plexus
T10
kidneys and ureters
PS: vagus nerve
S: thoracic and lumbar splanchnic nerves
T11, T12
Motor neurons[edit]
Main article: Motor neuron
Motor neurons of the autonomic nervous system are found in ‘’autonomic ganglia’’. Those of the parasympathetic branch are located close to the target organ whilst the ganglia of the sympathetic branch are located close to the spinal cord.
The sympathetic ganglia here, are found in two chains: the pre-vertebral and pre-aortic chains. The activity of autonomic ganglionic neurons is modulated by “preganglionic neurons” located in the central nervous system. Preganglionic sympathetic neurons are located in the spinal cord, at the thorax and upper lumbar levels. Preganglionic parasympathetic neurons are found in the medulla oblongata where they form visceral motor nuclei; the dorsal motor nucleus of the vagus nerve; the nucleus ambiguus, the salivatory nuclei, and in the sacral region of the spinal cord.
Function[edit]
Function of the autonomic nervous system [13]
Sympathetic and parasympathetic divisions typically function in opposition to each other. But this opposition is better termed complementary in nature rather than antagonistic. For an analogy, one may think of the sympathetic division as the accelerator and the parasympathetic division as the brake. The sympathetic division typically functions in actions requiring quick responses. The parasympathetic division functions with actions that do not require immediate reaction. The sympathetic system is often considered the "fight or flight" system, while the parasympathetic system is often considered the "rest and digest" or "feed and breed" system.
However, many instances of sympathetic and parasympathetic activity cannot be ascribed to "fight" or "rest" situations. For example, standing up from a reclining or sitting position would entail an unsustainable drop in blood pressure if not for a compensatory increase in the arterial sympathetic tonus. Another example is the constant, second-to-second, modulation of heart rate by sympathetic and parasympathetic influences, as a function of the respiratory cycles. In general, these two systems should be seen as permanently modulating vital functions, in usually antagonistic fashion, to achieve homeostasis. Some typical actions of the sympathetic and parasympathetic systems are listed below.
Sympathetic nervous system[edit]
Main article: Sympathetic nervous system
Promotes a fight-or-flight response, corresponds with arousal and energy generation, and inhibits digestion
Diverts blood flow away from the gastro-intestinal (GI) tract and skin via vasoconstriction
Blood flow to skeletal muscles and the lungs is enhanced (by as much as 1200% in the case of skeletal muscles)
Dilates bronchioles of the lung through circulating epinephrine, which allows for greater alveolar oxygen exchange
Increases heart rate and the contractility of cardiac cells (myocytes), thereby providing a mechanism for enhanced blood flow to skeletal muscles
Dilates pupils and relaxes the ciliary muscle to the lens, allowing more light to enter the eye and enhances far vision
Provides vasodilation for the coronary vessels of the heart
Constricts all the intestinal sphincters and the urinary sphincter
Inhibits peristalsis
Stimulates orgasm
Parasympathetic nervous system[edit]
Main article: Parasympathetic nervous system
The parasympathetic nervous system has been said to promote a "rest and digest" response, promotes calming of the nerves return to regular function, and enhancing digestion. Functions of nerves within the parasympathetic nervous system include:[citation needed]
Dilating blood vessels leading to the GI tract, increasing the blood flow.
Constricting the bronchiolar diameter when the need for oxygen has diminished
Dedicated cardiac branches of the vagus and thoracic spinal accessory nerves impart parasympathetic control of the heart (myocardium)
Constriction of the pupil and contraction of the ciliary muscles, facilitating accommodation and allowing for closer vision
Stimulating salivary gland secretion, and accelerates peristalsis, mediating digestion of food and, indirectly, the absorption of nutrients
Sexual. Nerves of the peripheral nervous system are involved in the erection of genital tissues via the pelvic splanchnic nerves 2–4. They are also responsible for stimulating sexual arousal.
Enteric nervous system[edit]
Main article: Enteric nervous system
The enteric nervous system is the intrinsic nervous system of the gastrointestinal system. It has been described as "the Second Brain of the Human Body".[14] Its functions include:
Sensing chemical and mechanical changes in the gut
Regulation secretions in the gut
Control of peristalsis and some other movements
Neurotransmitters[edit]
Main articles: Table of neurotransmitter actions in the ANS and Non-noradrenergic, non-cholinergic transmitter
A flow diagram showing the process of stimulation of adrenal medulla that makes it release adrenaline, that further acts on adrenoreceptors, indirectly mediating or mimicking sympathetic activity.
At the effector organs, sympathetic ganglionic neurons release noradrenaline (norepinephrine), along with other cotransmitters such as ATP, to act on adrenergic receptors, with the exception of the sweat glands and the adrenal medulla:
Acetylcholine is the preganglionic neurotransmitter for both divisions of the ANS, as well as the postganglionic neurotransmitter of parasympathetic neurons. Nerves that release acetylcholine are said to be cholinergic. In the parasympathetic system, ganglionic neurons use acetylcholine as a neurotransmitter to stimulate muscarinic receptors.
At the adrenal medulla, there is no postsynaptic neuron. Instead the presynaptic neuron releases acetylcholine to act on nicotinic receptors. Stimulation of the adrenal medulla releases adrenaline (epinephrine) into the bloodstream, which acts on adrenoceptors, thereby indirectly mediating or mimicking sympathetic activity.
A full table is found at Table of neurotransmitter actions in the ANS.
Caffeine effects[edit]
Caffeine is a bio-active ingredient found in commonly consumed beverages such as coffee, tea, and sodas. Short-term physiological effects of caffeine include increased blood pressure and sympathetic nerve outflow. Habitual consumption of caffeine may inhibit physiological short-term effects. Consumption of caffeinated espresso increases parasympathetic activity in habitual caffeine consumers; however, decaffeinated espresso inhibits parasympathetic activity in habitual caffeine consumers. It is possible that other bio-active ingredients in decaffeinated espresso may also contribute to the inhibition of parasympathetic activity in habitual caffeine consumers.[15]
Caffeine is capable of increasing work capacity while individuals perform strenuous tasks. In one study, caffeine provoked a greater maximum heart rate while a strenuous task was being performed compared to a placebo. This tendency is likely due to caffeine's ability to increase sympathetic nerve outflow. Furthermore, this study found that recovery after intense exercise was slower when caffeine was consumed prior to exercise. This finding is indicative of caffeine's tendency to inhibit parasympathetic activity in non-habitual consumers. The caffeine-stimulated increase in nerve activity is likely to evoke other physiological effects as the body attempts to maintain homeostasis.[16]
The effects of caffeine on parasympathetic activity may vary depending on the position of the individual when autonomic responses are measured. One study found that the seated position inhibited autonomic activity after caffeine consumption (75 mg); however, parasympathetic activity increased in the supine position. This finding may explain why some habitual caffeine consumers (75 mg or less) do not experience short-term effects of caffeine if their routine requires many hours in a seated position. It is important to note that the data supporting increased parasympathetic activity in the supine position was derived from an experiment involving participants between the ages of 25 and 30 who were considered healthy and sedentary. Caffeine may influence autonomic activity differently for individuals who are more active or elderly.[17]
See also[edit]
Dysautonomia
International Society for Autonomic Neuroscience
Lövheim cube of emotion
References[edit]
Jump up ^ "autonomic nervous system" at Dorland's Medical Dictionary
Jump up ^ Schmidt, A; Thews, G (1989). "Autonomic Nervous System". In Janig, W. Human Physiology (2 ed.). New York, NY: Springer-Verlag. pp. 333–370.
^ Jump up to: a b Allostatic load notebook: Parasympathetic Function - 1999, MacArthur research network, UCSF
Jump up ^ Langley, J.N. (1921). The Autonomic Nervous System Part 1. Cambridge: W. Heffer.
Jump up ^ Jänig, Wilfrid (2008). Integrative action of the autonomic nervous system : neurobiology of homeostasis (Digitally printed version. ed.). Cambridge: Cambridge University Press. p. 13. ISBN 978052106754-6.
Jump up ^ Furness, John (9 October 2007). "Enteric nervous system". Scholarpedia. doi:10.4249/scholarpedia.4064. Retrieved 8 October 2017.
Jump up ^ Willis, William D. (2004). "The Autonomic Nervous System and its central control". In Berne, Robert M. Physiology (5. ed.). St. Louis, Mo.: Mosby. ISBN 0323022251.
Jump up ^ Pocock, Gillian (2006). Human Physiology (3rd ed.). Oxford University Press. pp. 63–64. ISBN 978-0-19-856878-0.
Jump up ^ Belvisi, Maria G.; David Stretton, C.; Yacoub, Magdi; Barnes, Peter J. (1992). "Nitric oxide is the endogenous neurotransmitter of bronchodilator nerves in humans". European Journal of Pharmacology. 210 (2): 221–2. doi:10.1016/0014-2999(92)90676-U. PMID 1350993.
Jump up ^ Costanzo, Linda S. (2007). Physiology. Hagerstwon, MD: Lippincott Williams & Wilkins. p. 37. ISBN 0-7817-7311-3.
Jump up ^ Essential Clinical Anatomy. K.L. Moore & A.M. Agur. Lippincott, 2 ed. 2002. Page 199
^ Jump up to: a b Unless specified otherwise in the boxes, the source is: Moore, Keith L.; Agur, A. M. R. (2002). Essential Clinical Anatomy (2nd ed.). Lippincott Williams & Wilkins. p. 199. ISBN 978-0-7817-5940-3.
Jump up ^ Neil A. Campbell, Jane B. Reece: Biologie. Spektrum-Verlag Heidelberg-Berlin 2003, ISBN 3-8274-1352-4
Jump up ^ Hadhazy, Adam (February 12, 2010). "Think Twice: How the Gut's "Second Brain" Influences Mood and Well-Being". Scientific American.
Jump up ^ Zimmerman-Viehoff, Frank; Thayer, Julian; Koenig, Julian; Herrmann, Christian; Weber, Cora S.; Deter, Hans-Christian (May 1, 2016). "Short-term effects of espresso coffee on heart rate variability and blood pressure in habitual and non-habitual coffee consumers- a randomized crossover study". Nutritional Neuroscience. 19 (4): 169–175. Retrieved February 20, 2017.
Jump up ^ Bunsawat, Kanokwan; White, Daniel W; Kappus, Rebecca M; Baynard, Tracy (2015). "Caffeine delays autonomic recovery following acute exercise". European Journal of Preventive Cardiology. 22 (11): 1473–1479. Retrieved February 20, 2017.
Jump up ^ Monda, M.; Viggiano, An.; Vicidomini, C.; Viggiano, Al.; Iannaccone, T.; Tafuri, D.; De Luca, B. (2009). "Espresso coffee increases parasympathetic activity in young, healthy people". Nutritional Neuroscience. 12 (1): 43–48. Retrieved February 20, 2017.
External links[edit]
Autonomic nervous system article in Scholarpedia, by Ian Gibbins and Bill Blessing
ANS Medical Notes on rahulgladwin.com
NerveExpress is a fully automatic, non-invasive computer-based system designed for quantitative assessment of the autonomic nervous system (ANS) based on heart rate variability (HRV) analysis.
[show]
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e
Nervous system
Central nervous system
Meninges
Spinal cord
Brain
Rhombencephalon
Medulla
Pons
Cerebellum
Midbrain
Forebrain
Diencephalon
Retina
Optic nerve
Cerebrum
Limbic system
Peripheral nervous system
Somatic
Sensory nerve
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Cranial nerve
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Autonomic
Sympathetic
Parasympathetic
Enteric
[show]
v
t
e
Anatomy of the autonomic nervous system
Head
Sympathetic
Cervical ganglia: roots
long ciliary
Deep petrosal
Parasympathetic
Ciliary ganglion: roots
Sensory
Parasympathetic
Short ciliary
Pterygopalatine ganglion: deep petrosal
nerve of pterygoid canal
branches of distribution: greater palatine
inferior posterior nasal branches
lesser palatine
nasopalatine
medial superior posterior nasal branches
pharyngeal
Submandibular ganglion
Otic ganglion
Neck
Sympathetic
paravertebral ganglia: Cervical ganglia
Superior
Middle
Inferior
Stellate ganglion
prevertebral plexus: Cavernous plexus
Internal carotid
Chest
Sympathetic
paravertebral ganglia: Thoracic ganglia
prevertebral plexus: Cardiac plexus
Esophageal plexus
Pulmonary plexus
Thoracic aortic plexus
splanchnic nerves: cardiopulmonary
thoracic
cardiac nerves: Superior
Middle
Inferior
Abdomen
Sympathetic
paravertebral ganglia: Lumbar ganglia
prevertebral ganglia:
Celiac ganglia
Aorticorenal
Superior mesenteric ganglion
Inferior mesenteric ganglion
prevertebral plexus:
Celiac plexus
Hepatic
Splenic
Pancreatic
aorticorenal
Abdominal aortic plexus
Renal/Suprarenal
Superior mesenteric
Gastric
Inferior mesenteric
Spermatic
Ovarian
Superior hypogastric
hypogastric nerve
Superior rectal
Inferior hypogastric
Vesical
Prostatic / Cavernous nerves of penis
Uterovaginal
Middle rectal
splanchnic nerves: Lumbar splanchnic nerves
Enteric
Submucous plexus
Myenteric plexus
Pelvis
Sympathetic
paravertebral ganglia: Sacral ganglia
Ganglion impar
splanchnic nerves: Sacral splanchnic nerves
Parasympathetic
splanchnic nerves: Pelvic splanchnic nerves
Authority control
GND: 4062444-4
NDL: 00574849
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Geometric progression
From Wikipedia, the free encyclopedia
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Diagram illustrating three basic geometric sequences of the pattern 1(rn−1) up to 6 iterations deep. The first block is a unit block and the dashed line represents the infinite sum of the sequence, a number that it will forever approach but never touch: 2, 3/2, and 4/3 respectively.
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2.
Examples of a geometric sequence are powers rk of a fixed number r, such as 2k and 3k. The general form of a geometric sequence is
a , a r , a r 2 , a r 3 , a r 4 , … {\displaystyle a,\ ar,\ ar^{2},\ ar^{3},\ ar^{4},\ \ldots }
where r ≠ 0 is the common ratio and a is a scale factor, equal to the sequence's start value.
Contents
[hide]
1 Elementary properties
2 Geometric series
2.1 Derivation
2.2 Related formulas
2.3 Infinite geometric series
2.4 Complex numbers
3 Product
4 Relationship to geometry and Euclid's work
5 See also
6 References
7 External links
Elementary properties[edit]
The n-th term of a geometric sequence with initial value a and common ratio r is given by
a n = a r n − 1 . {\displaystyle a_{n}=a\,r^{n-1}.}
Such a geometric sequence also follows the recursive relation
a n = r a n − 1 {\displaystyle a_{n}=r\,a_{n-1}} for every integer n ≥ 1. {\displaystyle n\geq 1.}
Generally, to check whether a given sequence is geometric, one simply checks whether successive entries in the sequence all have the same ratio.
The common ratio of a geometric sequence may be negative, resulting in an alternating sequence, with numbers switching from positive to negative and back. For instance
1, −3, 9, −27, 81, −243, ...
is a geometric sequence with common ratio −3.
The behaviour of a geometric sequence depends on the value of the common ratio.
If the common ratio is:
Positive, the terms will all be the same sign as the initial term.
Negative, the terms will alternate between positive and negative.
Greater than 1, there will be exponential growth towards positive or negative infinity (depending on the sign of the initial term).
1, the progression is a constant sequence.
Between −1 and 1 but not zero, there will be exponential decay towards zero.
−1, the progression is an alternating sequence
Less than −1, for the absolute values there is exponential growth towards (unsigned) infinity, due to the alternating sign.
Geometric sequences (with common ratio not equal to −1, 1 or 0) show exponential growth or exponential decay, as opposed to the linear growth (or decline) of an arithmetic progression such as 4, 15, 26, 37, 48, … (with common difference 11). This result was taken by T.R. Malthus as the mathematical foundation of his Principle of Population. Note that the two kinds of progression are related: exponentiating each term of an arithmetic progression yields a geometric progression, while taking the logarithm of each term in a geometric progression with a positive common ratio yields an arithmetic progression.
An interesting result of the definition of a geometric progression is that for any value of the common ratio, any three consecutive terms a, b and c will satisfy the following equation:
b 2 = a c {\displaystyle b^{2}=ac}
where b is considered to be the geometric mean between a and c.
Geometric series[edit]
This section may stray from the topic of the article into the topic of another article, Geometric series. Please help improve this section or discuss this issue on the talk page. (February 2014)
2 + 10 + 50 + 250 = 312
− ( 10 + 50 + 250 + 1250 = 5 × 312 )
2 − 1250 = (1 − 5) × 312
Computation of the sum 2 + 10 + 50 + 250. The sequence is multiplied term by term by 5, and then subtracted from the original sequence. Two terms remain: the first term, a, and the term one beyond the last, or arm. The desired result, 312, is found by subtracting these two terms and dividing by 1 − 5.
A geometric series is the sum of the numbers in a geometric progression. For example:
2 + 10 + 50 + 250 = 2 + 2 × 5 + 2 × 5 2 + 2 × 5 3 . {\displaystyle 2+10+50+250=2+2\times 5+2\times 5^{2}+2\times 5^{3}.}
Letting a be the first term (here 2), n be the number of terms (here 4), and r be the constant that each term is multiplied by to get the next term (here 5), the sum is given by:
a ( 1 − r n ) 1 − r {\displaystyle {\frac {a(1-r^{n})}{1-r}}}
In the example above, this gives:
2 + 10 + 50 + 250 = 2 ( 1 − 5 4 ) 1 − 5 = − 1248 − 4 = 312. {\displaystyle 2+10+50+250={\frac {2(1-5^{4})}{1-5}}={\frac {-1248}{-4}}=312.}
The formula works for any real numbers a and r (except r = 1, which results in a division by zero). For example:
− 2 π + 4 π 2 − 8 π 3 = − 2 π + ( − 2 π ) 2 + ( − 2 π ) 3 = − 2 π ( 1 − ( − 2 π ) 3 ) 1 − ( − 2 π ) = − 2 π ( 1 + 8 π 3 ) 1 + 2 π ≈ − 214.855. {\displaystyle -2\pi +4\pi ^{2}-8\pi ^{3}=-2\pi +(-2\pi )^{2}+(-2\pi )^{3}={\frac {-2\pi (1-(-2\pi )^{3})}{1-(-2\pi )}}={\frac {-2\pi (1+8\pi ^{3})}{1+2\pi }}\approx -214.855.}
Since the derivation (below) does not depend on a and r being real, it holds for complex numbers as well.
Derivation[edit]
To derive this formula, first write a general geometric series as:
∑ k = 1 n a r k − 1 = a r 0 + a r 1 + a r 2 + a r 3 + ⋯ + a r n − 1 . {\displaystyle \sum _{k=1}^{n}ar^{k-1}=ar^{0}+ar^{1}+ar^{2}+ar^{3}+\cdots +ar^{n-1}.}
We can find a simpler formula for this sum by multiplying both sides of the above equation by 1 − r, and we'll see that
( 1 − r ) ∑ k = 1 n a r k − 1 = ( 1 − r ) ( a r 0 + a r 1 + a r 2 + a r 3 + ⋯ + a r n − 1 ) = a r 0 + a r 1 + a r 2 + a r 3 + ⋯ + a r n − 1 − a r 1 − a r 2 − a r 3 − ⋯ − a r n − 1 − a r n = a − a r n {\displaystyle {\begin{aligned}(1-r)\sum _{k=1}^{n}ar^{k-1}&=(1-r)(ar^{0}+ar^{1}+ar^{2}+ar^{3}+\cdots +ar^{n-1})\\&=ar^{0}+ar^{1}+ar^{2}+ar^{3}+\cdots +ar^{n-1}-ar^{1}-ar^{2}-ar^{3}-\cdots -ar^{n-1}-ar^{n}\\&=a-ar^{n}\end{aligned}}}
since all the other terms cancel. If r ≠ 1, we can rearrange the above to get the convenient formula for a geometric series that computes the sum of n terms:
∑ k = 1 n a r k − 1 = a ( 1 − r n ) 1 − r . {\displaystyle \sum _{k=1}^{n}ar^{k-1}={\frac {a(1-r^{n})}{1-r}}.}
Related formulas[edit]
If one were to begin the sum not from k=1, but from a different value, say m, then
∑ k = m n a r k = a ( r m − r n + 1 ) 1 − r , {\displaystyle \sum _{k=m}^{n}ar^{k}={\frac {a(r^{m}-r^{n+1})}{1-r}},}
provided r ≠ 1 {\displaystyle r\neq 1} and a ( n − m + 1 ) {\displaystyle a(n-m+1)} when r = 1 {\displaystyle r=1} .
Differentiating this formula with respect to r allows us to arrive at formulae for sums of the form
G s ( n , r ) := ∑ k = 0 n k s r k . {\displaystyle G_{s}(n,r):=\sum _{k=0}^{n}k^{s}r^{k}.}
For example:
d d r ∑ k = 0 n r k = ∑ k = 1 n k r k − 1 = 1 − r n + 1 ( 1 − r ) 2 − ( n + 1 ) r n 1 − r . {\displaystyle {\frac {d}{dr}}\sum _{k=0}^{n}r^{k}=\sum _{k=1}^{n}kr^{k-1}={\frac {1-r^{n+1}}{(1-r)^{2}}}-{\frac {(n+1)r^{n}}{1-r}}.}
For a geometric series containing only even powers of r multiply by 1 − r2 :
( 1 − r 2 ) ∑ k = 0 n a r 2 k = a − a r 2 n + 2 . {\displaystyle (1-r^{2})\sum _{k=0}^{n}ar^{2k}=a-ar^{2n+2}.}
Then
∑ k = 0 n a r 2 k = a ( 1 − r 2 n + 2 ) 1 − r 2 . {\displaystyle \sum _{k=0}^{n}ar^{2k}={\frac {a(1-r^{2n+2})}{1-r^{2}}}.}
Equivalently, take r2 as the common ratio and use the standard formulation.
For a series with only odd powers of r
( 1 − r 2 ) ∑ k = 0 n a r 2 k + 1 = a r − a r 2 n + 3 {\displaystyle (1-r^{2})\sum _{k=0}^{n}ar^{2k+1}=ar-ar^{2n+3}}
and
∑ k = 0 n a r 2 k + 1 = a r ( 1 − r 2 n + 2 ) 1 − r 2 . {\displaystyle \sum _{k=0}^{n}ar^{2k+1}={\frac {ar(1-r^{2n+2})}{1-r^{2}}}.}
An exact formula for the generalized sum G s ( n , r ) {\displaystyle G_{s}(n,r)} when s ∈ N {\displaystyle s\in \mathbb {N} } is expanded by the Stirling numbers of the second kind as [1]
G s ( n , r ) = ∑ j = 0 s { s j } x j d j d x j [ 1 − x n + 1 1 − x ] . {\displaystyle G_{s}(n,r)=\sum _{j=0}^{s}\left\lbrace {s \atop j}\right\rbrace x^{j}{\frac {d^{j}}{dx^{j}}}\left[{\frac {1-x^{n+1}}{1-x}}\right].}
Infinite geometric series[edit]
This section may stray from the topic of the article into the topic of another article, Geometric series. Please help improve this section or discuss this issue on the talk page. (February 2014)
Main article: Geometric series
An infinite geometric series is an infinite series whose successive terms have a common ratio. Such a series converges if and only if the absolute value of the common ratio is less than one (|r| < 1). Its value can then be computed from the finite sum formula
∑ k = 0 ∞ a r k = lim n → ∞ ∑ k = 0 n a r k = lim n → ∞ a ( 1 − r n + 1 ) 1 − r = a 1 − r − lim n → ∞ a r n + 1 1 − r {\displaystyle \sum _{k=0}^{\infty }ar^{k}=\lim _{n\to \infty }{\sum _{k=0}^{n}ar^{k}}=\lim _{n\to \infty }{\frac {a(1-r^{n+1})}{1-r}}={\frac {a}{1-r}}-\lim _{n\to \infty }{\frac {ar^{n+1}}{1-r}}}
Animation, showing convergence of partial sums of geometric progression ∑ k = 0 n q k {\displaystyle \sum \limits _{k=0}^{n}q^{k}} (red line) to its sum 1 1 − q {\displaystyle {1 \over 1-q}} (blue line) for | q | < 1 {\displaystyle |q|<1} .
Diagram showing the geometric series 1 + 1/2 + 1/4 + 1/8 + ⋯ which converges to 2.
Since:
r n + 1 → 0 as n → ∞ when | r | < 1. {\displaystyle r^{n+1}\to 0{\mbox{ as }}n\to \infty {\mbox{ when }}|r|<1.}
Then:
∑ k = 0 ∞ a r k = a 1 − r − 0 = a 1 − r {\displaystyle \sum _{k=0}^{\infty }ar^{k}={\frac {a}{1-r}}-0={\frac {a}{1-r}}}
For a series containing only even powers of r {\displaystyle r} ,
∑ k = 0 ∞ a r 2 k = a 1 − r 2 {\displaystyle \sum _{k=0}^{\infty }ar^{2k}={\frac {a}{1-r^{2}}}}
and for odd powers only,
∑ k = 0 ∞ a r 2 k + 1 = a r 1 − r 2 {\displaystyle \sum _{k=0}^{\infty }ar^{2k+1}={\frac {ar}{1-r^{2}}}}
In cases where the sum does not start at k = 0,
∑ k = m ∞ a r k = a r m 1 − r {\displaystyle \sum _{k=m}^{\infty }ar^{k}={\frac {ar^{m}}{1-r}}}
The formulae given above are valid only for |r| < 1. The latter formula is valid in every Banach algebra, as long as the norm of r is less than one, and also in the field of p-adic numbers if |r|p < 1. As in the case for a finite sum, we can differentiate to calculate formulae for related sums. For example,
d d r ∑ k = 0 ∞ r k = ∑ k = 1 ∞ k r k − 1 = 1 ( 1 − r ) 2 {\displaystyle {\frac {d}{dr}}\sum _{k=0}^{\infty }r^{k}=\sum _{k=1}^{\infty }kr^{k-1}={\frac {1}{(1-r)^{2}}}}
This formula only works for |r| < 1 as well. From this, it follows that, for |r| < 1,
∑ k = 0 ∞ k r k = r ( 1 − r ) 2 ; ∑ k = 0 ∞ k 2 r k = r ( 1 + r ) ( 1 − r ) 3 ; ∑ k = 0 ∞ k 3 r k = r ( 1 + 4 r + r 2 ) ( 1 − r ) 4 {\displaystyle \sum _{k=0}^{\infty }kr^{k}={\frac {r}{\left(1-r\right)^{2}}}\,;\,\sum _{k=0}^{\infty }k^{2}r^{k}={\frac {r\left(1+r\right)}{\left(1-r\right)^{3}}}\,;\,\sum _{k=0}^{\infty }k^{3}r^{k}={\frac {r\left(1+4r+r^{2}\right)}{\left(1-r\right)^{4}}}}
Also, the infinite series 1/2 + 1/4 + 1/8 + 1/16 + ⋯ is an elementary example of a series that converges absolutely.
It is a geometric series whose first term is 1/2 and whose common ratio is 1/2, so its sum is
1 2 + 1 4 + 1 8 + 1 16 + ⋯ = 1 / 2 1 − ( + 1 / 2 ) = 1. {\displaystyle {\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots ={\frac {1/2}{1-(+1/2)}}=1.}
The inverse of the above series is 1/2 − 1/4 + 1/8 − 1/16 + ⋯ is a simple example of an alternating series that converges absolutely.
It is a geometric series whose first term is 1/2 and whose common ratio is −1/2, so its sum is
1 2 − 1 4 + 1 8 − 1 16 + ⋯ = 1 / 2 1 − ( − 1 / 2 ) = 1 3 . {\displaystyle {\frac {1}{2}}-{\frac {1}{4}}+{\frac {1}{8}}-{\frac {1}{16}}+\cdots ={\frac {1/2}{1-(-1/2)}}={\frac {1}{3}}.}
Complex numbers[edit]
The summation formula for geometric series remains valid even when the common ratio is a complex number. In this case the condition that the absolute value of r be less than 1 becomes that the modulus of r be less than 1. It is possible to calculate the sums of some non-obvious geometric series. For example, consider the proposition
∑ k = 0 ∞ sin  ( k x ) r k = r sin  ( x ) 1 + r 2 − 2 r cos  ( x ) {\displaystyle \sum _{k=0}^{\infty }{\frac {\sin(kx)}{r^{k}}}={\frac {r\sin(x)}{1+r^{2}-2r\cos(x)}}}
The proof of this comes from the fact that
sin  ( k x ) = e i k x − e − i k x 2 i , {\displaystyle \sin(kx)={\frac {e^{ikx}-e^{-ikx}}{2i}},}
which is a consequence of Euler's formula. Substituting this into the original series gives
∑ k = 0 ∞ sin  ( k x ) r k = 1 2 i [ ∑ k = 0 ∞ ( e i x r ) k − ∑ k = 0 ∞ ( e − i x r ) k ] {\displaystyle \sum _{k=0}^{\infty }{\frac {\sin(kx)}{r^{k}}}={\frac {1}{2i}}\left[\sum _{k=0}^{\infty }\left({\frac {e^{ix}}{r}}\right)^{k}-\sum _{k=0}^{\infty }\left({\frac {e^{-ix}}{r}}\right)^{k}\right]} .
This is the difference of two geometric series, and so it is a straightforward application of the formula for infinite geometric series that completes the proof.
Product[edit]
The product of a geometric progression is the product of all terms. If all terms are positive, then it can be quickly computed by taking the geometric mean of the progression's first and last term, and raising that mean to the power given by the number of terms. (This is very similar to the formula for the sum of terms of an arithmetic sequence: take the arithmetic mean of the first and last term and multiply with the number of terms.)
∏ i = 0 n a r i = ( a 0 ⋅ a n ) n + 1 {\displaystyle \prod _{i=0}^{n}ar^{i}=\left({\sqrt {a_{0}\cdot a_{n}}}\right)^{n+1}} (if a , r > 0 {\displaystyle a,r>0} ).
Proof:
Let the product be represented by P:
P = a ⋅ a r ⋅ a r 2 ⋯ a r n − 1 ⋅ a r n {\displaystyle P=a\cdot ar\cdot ar^{2}\cdots ar^{n-1}\cdot ar^{n}} .
Now, carrying out the multiplications, we conclude that
P = a n + 1 r 1 + 2 + 3 + ⋯ + ( n − 1 ) + ( n ) {\displaystyle P=a^{n+1}r^{1+2+3+\cdots +(n-1)+(n)}} .
Applying the sum of arithmetic series, the expression will yield
P = a n + 1 r n ( n + 1 ) 2 {\displaystyle P=a^{n+1}r^{\frac {n(n+1)}{2}}} .
P = ( a r n 2 ) n + 1 {\displaystyle P=(ar^{\frac {n}{2}})^{n+1}} .
We raise both sides to the second power:
P 2 = ( a 2 r n ) n + 1 = ( a ⋅ a r n ) n + 1 {\displaystyle P^{2}=(a^{2}r^{n})^{n+1}=(a\cdot ar^{n})^{n+1}} .
Consequently,
P 2 = ( a 0 ⋅ a n ) n + 1 {\displaystyle P^{2}=(a_{0}\cdot a_{n})^{n+1}} and
P = ( a 0 ⋅ a n ) n + 1 2 {\displaystyle P=(a_{0}\cdot a_{n})^{\frac {n+1}{2}}} ,
which concludes the proof.
Relationship to geometry and Euclid's work[edit]
Books VIII and IX of Euclid's Elements analyzes geometric progressions (such as the powers of two, see the article for details) and give several of their properties.[2]
See also[edit]
Arithmetic progression
Arithmetico-geometric sequence
Linear difference equation
Exponential function
Harmonic progression
Harmonic series
Infinite series
Preferred number
Thomas Robert Malthus
Geometric distribution
References[edit]
Jump up ^ "Set Partitions: Stirling Numbers". Digital Library of Mathematical Functions. Retrieved 24 May 2018.
Jump up ^ * Heath, Thomas L. (1956). The Thirteen Books of Euclid's Elements (2nd ed. [Facsimile. Original publication: Cambridge University Press, 1925] ed.). New York: Dover Publications.
Hall & Knight, Higher Algebra, p. 39, ISBN 81-8116-000-2
External links[edit]
Hazewinkel, Michiel, ed. (2001) [1994], "Geometric progression", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Derivation of formulas for sum of finite and infinite geometric progression at Mathalino.com
Geometric Progression Calculator
Nice Proof of a Geometric Progression Sum at sputsoft.com
Weisstein, Eric W. "Geometric Series". MathWorld.
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