Infinite series Definition and 390 Threads

In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics) through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance.
For a long time, the idea that such a potentially infinite summation could produce a finite result was considered paradoxical. This paradox was resolved using the concept of a limit during the 17th century. Zeno's paradox of Achilles and the tortoise illustrates this counterintuitive property of infinite sums: Achilles runs after a tortoise, but when he reaches the position of the tortoise at the beginning of the race, the tortoise has reached a second position; when he reaches this second position, the tortoise is at a third position, and so on. Zeno concluded that Achilles could never reach the tortoise, and thus that movement does not exist. Zeno divided the race into infinitely many sub-races, each requiring a finite amount of time, so that the total time for Achilles to catch the tortoise is given by a series. The resolution of the paradox is that, although the series has an infinite number of terms, it has a finite sum, which gives the time necessary for Achilles to catch up with the tortoise.
In modern terminology, any (ordered) infinite sequence



(

a

1


,

a

2


,

a

3


,

)


{\displaystyle (a_{1},a_{2},a_{3},\ldots )}
of terms (that is, numbers, functions, or anything that can be added) defines a series, which is the operation of adding the ai one after the other. To emphasize that there are an infinite number of terms, a series may be called an infinite series. Such a series is represented (or denoted) by an expression like

or, using the summation sign,

The infinite sequence of additions implied by a series cannot be effectively carried on (at least in a finite amount of time). However, if the set to which the terms and their finite sums belong has a notion of limit, it is sometimes possible to assign a value to a series, called the sum of the series. This value is the limit as n tends to infinity (if the limit exists) of the finite sums of the n first terms of the series, which are called the nth partial sums of the series. That is,
When this limit exists, one says that the series is convergent or summable, or that the sequence



(

a

1


,

a

2


,

a

3


,

)


{\displaystyle (a_{1},a_{2},a_{3},\ldots )}
is summable. In this case, the limit is called the sum of the series. Otherwise, the series is said to be divergent.The notation






i
=
1






a

i




{\textstyle \sum _{i=1}^{\infty }a_{i}}
denotes both the series—that is the implicit process of adding the terms one after the other indefinitely—and, if the series is convergent, the sum of the series—the result of the process. This is a generalization of the similar convention of denoting by



a
+
b


{\displaystyle a+b}
both the addition—the process of adding—and its result—the sum of a and b.
Generally, the terms of a series come from a ring, often the field




R



{\displaystyle \mathbb {R} }
of the real numbers or the field




C



{\displaystyle \mathbb {C} }
of the complex numbers. In this case, the set of all series is itself a ring (and even an associative algebra), in which the addition consists of adding the series term by term, and the multiplication is the Cauchy product.

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  1. R

    Infinite Series: U(n+1)/U(n) Calculation

    The given series is: 1+[(a+1)/(b+1)]+[(a+1)(2*a+1)/(b+1)(2*b+1)]+[(a+1)(2*a+1)(3*a+1)/(b+1)(2*b+1)(3*b+1)]+...∞ Problem: To find U(n+1)/U(n). My approach: Removing the first term(1) of the series and making the second term the first,third term the second and so on... I get...
  2. B

    Where Can I Practice Integrals Using Infinite Series?

    On the bottom of page 24 & top of page 25 of this pdf an integral is beautifully computed by breaking it up into an infinite series. Is there any reference where I could get practice in working integrals like these?
  3. P

    Convergence or Divergence of Infinite Series: Methods and Examples

    Homework Statement Determine convergence or divergence using any method covered so far*: Ʃ(1/(n*ln(n)^2 - n)) from n = 1 to infinity*The methods are the following: - Dichotomy for positive series (if the partial sums are bounded above and the series is positive, the series converges) -...
  4. M

    Infinite Series Homework: Determine Convergence/Divergence

    Homework Statement Determine whether the series diverges or converges. (1+2) / (1+3)+ ((1+2+4)/(1+3+9))+ ((1+2+4+8)/(1+3+9+27)) + ...The Attempt at a Solution I have split up the series into two (denominator and numerator): an = (1+2) + (1+2+4) + (1+2+4+8)+... = (1)n + 2n + 4(n-1) + ... bn...
  5. E

    How Do You Solve This Arithmetico-Geometric Series?

    Trying to solve this infinite series?? Hey folks! I've spent hours trying to solve this and have exhausted all available resources.. I just need to be pointed in the right direction! Homework Statement Compute the sum of the infinite series (I believe this is an arithmetico geometric...
  6. Seydlitz

    Proof of the preliminary test for infinite series

    Homework Statement Preliminary test: If the terms of an infinite series do not tend to zero, the series diverges. In other words if ##\lim_{ n \to \infty}a_n \neq 0## then the series diverges. But if the limit is 0 we have to test further. Suppose a series a series satisfy this condition...
  7. jacobi1

    MHB Sum of Cosines: Find the Infinite Series

    Find \sum_{n=0}^\infty \frac{\cos(nx)}{2^n}.
  8. Saitama

    Infinite series - Inverse trigonometry

    Homework Statement The sum of the infinite terms of the series \text{arccot}\left(1^2+\frac{3}{4}\right)+\text{arccot}\left(2^2+\frac{3}{4}\right)+\text{arccot}\left(3^2+\frac{3}{4}\right)+... is equal to A)arctan(1) B)arctan(2) C)arctan(3) D)arctan(4) Ans: B Homework Equations The Attempt at...
  9. jeffer vitola

    MHB Solving the Limit of an Infinite Series

    hello ... I propose this exercise for you to solve on various methods ...\[\lim_{n \to{+}\infty}{\frac{1}{n}\sum_{i=1}^n({1+\frac{i}{n}}})^{-2}\]thanks att jefferson alexander vitola(Smile)
  10. M

    Analytical solution of an infinite series

    How to find the value of an infinite series. for e.g.Ʃ_{n=1}^{\infty} (β^{n-1}y^{R^{n}}e^{A(1-R^{2n})}) where β<1, R<1, y>1, and A>0? Note that this series is covergent by Ratio test. I already have the numerical solution of the above. However, I am interested in analytical solution...
  11. S

    Why is the sum of 1/(n2^n) from 1 to infinity equal to log 2?

    I was looking at this topic: http://mathoverflow.net/questions/17960/google-question-in-a-country-in-which-people-only-want-boys-closed And the top answer uses the fact that the sum from 1 to infinity of 1/(x2^x) is log 2. Why is this true? Thanks in advance.
  12. AGNuke

    Summing Infinite Series: A Shortcut Using Differentiation

    Given S, an Infinite Series Summation, find \frac{1728}{485}S S=1^2+\frac{3^2}{5^2}+\frac{5^2}{5^4}+\frac{7^2}{5^6}+... I found out the formula for (r+1)th term of the series, hence making the series asS=1+\sum_{r=1}^{\infty}\frac{(2r+1)^2}{(5^r)^2} Now I have a hard time guessing what to do...
  13. D

    Find the sum of the infinite series

    Find the series sum ln2/2 – ln3/3 + ln4/4 – ln5/5 + ….
  14. S

    MHB Summing Infinite Series with Dilogarithms

    Hi everyone ;) I have a challenging problem which I would like to share with you. Prove that \[\frac{1}{2^2}+ \frac{1}{3^2} \left(1+\frac{1}{2} \right)^2+\frac{1}{4^2} \left( 1+\frac{1}{2} +\frac{1}{3}\right)^2 + \frac{1}{5^2} \left( 1+\frac{1}{2} +\frac{1}{3}+\frac{1}{4}\right)^2 +\cdots=...
  15. T

    Summing an infinite series question

    Homework Statement I need to fin the sum of the following two infinite series: 1. Ʃ[n=0 to ∞] ((2^n + 3^n)/6^n) and 2. Ʃ[n=2 to ∞] (2^n + (3^n / n^2)) (1/3^n) Homework Equations use the sum Ʃ[n=2 to ∞] (1/n^2) = ∏^2 / 6 as necessary The Attempt at a Solution I tried to manipulate them...
  16. T

    Comparison Test problem with infinite series

    Homework Statement I need to use the Comparison Test or the Limit Comparison Test to determine whether or not this series converges: ∑ sin(1/n^2) from 1 to ∞ Homework Equations Limit Comparison Test: Let {An} and {Bn} be positive sequences. Assume the following limit exists: L =...
  17. T

    MHB Which Tests Determine Convergence for These Series?

    Test these for convergence. 5. infinity E...((n!)^2((2n)!)^2)/((n^2 + 2n)!(n + 1)!) n = 0 6. infinity E...(1 - e ^ -((n^2 + 3n))/n)/(n^2) n = 3 note: for #3: -((n^2 + 3n))/n) is all to the power of e Btw, E means sum. Which tests should I use to solve these?
  18. T

    MHB Which tests should I use for convergence?

    Test these for convergence. 3. infinity E...((-1)^n)*(n^3 + 3n)/((n^2) + 7n) n = 2 4. infinity E...ln(n^3)/n^2 n = 2 note: for #3: -((n^2 + 3n))/n) is all to the power of e Btw, E means sum. Which tests should I use to solve these?
  19. T

    MHB Do These Infinite Series Converge?

    Test these for convergence. 1. infinity E...n!/(n! + 3^n) n = 0 2. infinity E...(n - (1/n))^-n n = 1 Btw, E means sum. Which tests should I use to solve these?
  20. K

    Evaluate the indefinite integral as an infinite series

    Homework Statement Evaluate the indefinite integral as an infinite series ∫ sin(x2) dx Homework Equations The Macluarin series of sin x = ∞ Ʃ (-1)nx2n+1/(2n+1)! n=0 The Macluarin series for sin(x2) = ∞ Ʃ (-1)x4n+2/(2n+1)! n=0 The Attempt...
  21. M

    Can the Infinite Series Theorem Extend to Negative Indices?

    For the series such that: \Sigma _{n=1} ^{\infty} a_n =\Sigma _{n=1} ^{\infty} b_n A certain theorem says that these series are equal even if a_n = b_n only for n>m. That is, even if two infinite series differ for a finite number of terms, it will still converge for the same sum. I am thinking...
  22. B

    Infinite Series with log natural Question

    Homework Statement ∞ Ʃ (-1)^(k+1) / kln(k) k=2 Homework Equations integral test, p test, comparison test, limit comparison, ratio test, root test. The Attempt at a Solution In class so far we have not learned the alternating series test so i can't use that test. So far I have...
  23. J

    MHB Sum of Infinite Series: $y^2+2y$

    If $\displaystyle y=\frac{3}{4}+\frac{3*5}{4*8}+\frac{3*5*7}{4*8*12}+...\infty$. Then $y^2+2y = $
  24. P

    How to make a surface plot involving an infinite series

    How to make a surface plot involving an infinite series in Matlab Solving Laplace's equation for electric potential for a 2D surface yields: V(x,y) = 4 Vo/pi * Ʃ (n=1,3,5,...) (1/n e^(-npi*x/a) sin(n*pi*y/a) ,where a is and Vo are constants ...it's convoluted, but basically, I need to...
  25. S

    What Is the Radius and Interval of Convergence for This Series?

    Homework Statement Find the radius of convergence and interval of convergence for the following infinite series \sum_{n=1}^{\∞} \frac{x^n n^2}{3 \cdot 6 \cdot 9 \cdot ... (3n)} Homework Equations Ratio test The Attempt at a Solution Using ratio test we get im not sure how to...
  26. P

    Estimated Sums in Infinite Series Problem

    Homework Statement How many terms of the series do we need to add in order to find the sum to the indicated accuracy? Ʃ((-1)n)/(n(10n)) from n=1 to infinity |error| <.0001 I keep ending up with n=log(4)-log(n)
  27. S

    So, the infinite series converges for a>2 and diverges for a=2.

    Homework Statement Show that the infinite series \sum_{n=0}^{\infty} (\sqrt{n^a+1}-\sqrt{n^a}) Converges for a>2 and diverges for x =2The Attempt at a SolutionI'm reviewing series, which I studied a certain time ago and picking some questions at random, I can't solve this one. I tried every...
  28. P

    MHB Strange inequality of infinite series

    Hi everybody, while doing a complex analysis exercise, i came to a strange inequality which i don't know how to interpretate. Suppose you have a sequence $\{a_j\}$ of positive real number. Let $\rho$ a positive real number. The inequality i found after some calculation is...
  29. U

    Find the sum to infinite series

    Homework Statement cot^-1 3 + cot^-1 7 + cot^-1 13+... Homework Equations The Attempt at a Solution I first tried to write the nth term of the series t_n = cot^{-1}\left( 2^n + (2n-1) \right) Then I tried to calculate the limit as n→∞. But I simply can't do that. I mean I...
  30. A

    Proving the Summation of an Infinite Series

    1. Homework Statement ∑ i=1 to n1+(1/i2)+(1/(1+i)2)−−−−−−−−−−−−−−−−−−−−√ = n(n+2)/n+1 2. The attempt at a solution First I did the base case of p(1) showing 3/2 on the LHS equals the 3/2 on the RHS. Then I assumed p(k) and wrote out the formula with k in it. Then prove p(k+1)= p(k)+...
  31. fluidistic

    Infinite series, probably related to Fourier transform?

    Homework Statement A function f(x) has the following series expansion: ##f(x)=\sum _{n=0}^\infty \frac{c_n x^n}{n!}##. Write down the function ##g(y)=\sum _{n=0}^\infty c_n y^n## under a closed form in function of f(x). Homework Equations Not sure at all. The Attempt at a Solution...
  32. D

    MHB Sum of an Infinite Series with Real Exponent p

    $\sum\limits_{n = 2}^{\infty}n^p\left(\frac{1}{\sqrt{n - 1}} - \frac{1}{\sqrt{n}}\right)$ where p is any fixed real number. If this was just the telescoping series or the p-series, this wouldn't be a problem.
  33. D

    Choosing a test for infinite series.

    Homework Statement The problem is part of a review and we are only to determine if the series converges or diverges by any test, and state the test. ##\sum_{n=1}^\infty(\frac{k}{k+1})^k## My work so far I know that the root test gives an inconclusive answer and from there I moved...
  34. A

    Solve Sum of Infinite Series: cos(n*pi)/5^n

    Question says: \sum(cos(n*pi)/5^n) from 0 to infinity. Proved that it converges: ratio test goes to abs(cos(pi*(n+1))/5cos(pi*n)) with some basic algebra. As n goes to infinity, this approaches -1/5 (absolute value giving 1/5) since cos(pi*(n+1))/cos(pi*n) is always -1, excepting the...
  35. B

    Equivalence of Two Infinite Series

    I am having a difficult time seeing how \sum_{n=0}^{\infty} ((-1)^n + 1)x^n is equivalent to 2\sum_{n=0}^{\infty} x^{2n}
  36. B

    Proving Euler's Formula using infinite series.

    Homework Statement I need to show that both sin(x) and cos(x) are absolutely convergent. Here's my work so far, Theorem: ℯix = cos(x) + i*sin(x) (1) Proof: This...
  37. B

    Can we Prove \lim_{n→∞}S_{n-1} = L Given \lim_{n→∞}S_{n} = L?

    This might sound like a dumb question, but it's actually not too obvious to me. If we know that \lim_{n→∞}S_{n} = L , can we prove that \lim_{n→∞}S_{n-1} = L ? I'm actually using this as a lemma in one of my other proofs (the proof that the nth term of a convergent sum approaches 0), but...
  38. P

    What is the Limit of a Sequence with a Common Ratio of 1/2?

    S = \frac{1}{2} + \frac{1}{4} + ... + (\frac{1}{2^n}) I noticed that this is a sum of a infinite series with the common ratio being 1/2, so using \frac{1}{1-1/2} I get S = 2, however with this question there is a hint saying multiply S by 2, which I did not use so I'm worrying if I done...
  39. B

    Determining Convergence/Divergence Of Infinite Series

    Homework Statement I attached the solution to the problem. Homework Equations The Attempt at a Solution I can see that the infinite series diverges by looking at a few terms, but how would I find a general term for the infinite series, to evaluate it analytically?
  40. W

    Infinite Series using Falling Factorials

    Homework Statement Determine \sum_{k=0}^\infty \frac{1}{(4k+1)(4k+2)(4k+3)(4k+4)}. Homework Equations ## (x)_m=x(x-1)(x-2)...(x-(m-1)) ##, integer ##m\geq0##. ## (x)_{-m}=\frac{1}{(x+1)(x+2)...(x+m)}##, integer ##m>0##. ## Δ((x)_m)=m(x)_{m-1}## \sum_{a\leq k<b}...
  41. B

    Truncated form of a infinite series

    Hi, In griffith's "Introduction to Electrodynamics" he indicates that a specific infinite series has a truncated form (the series and truncated form are given below) And he says the reader can try to show that it indeed has that form...
  42. Y

    Exploring Infinite Series: Solving \sum_{n=1}^{\infty}\frac{n}{k^{n}}

    Recently I have been playing around with infinite series and related topics, when I realized I couldn't figure out how to solve something of the form \sum_{n=1}^{\infty}\frac{n}{k^{n}} How would you go about finding a sum like this?
  43. S

    How Can I Deepen My Understanding of Infinite Series and Sequences?

    Hello. Having already learned about infinite series and sequences in my calculus class, I'm quite interested in them and especially in learning more about them. If any of you have in mind any good books on the subject which you can recommend to me, it will be very much appreciated...
  44. H

    How can I calculate the sum of the following infinite series?

    Homework Statement 1) \displaystyle \sum\limits_{n=1}^{\infty }{\frac{{{n}^{3}}}{{{e}^{n}}}} 2) \displaystyle \sum\limits_{n=1}^{\infty }{\frac{-1}{2\left( n+1 \right)}+\frac{2}{\left( n+2 \right)}-\frac{3}{n\left( n+3 \right)}}Homework Equations The Attempt at a Solution I've proved that...
  45. C

    Infinite series convergence question:

    Homework Statement Does \sum _{ n=1 }^{ \infty }{ \frac { { \alpha }^{ n }{ n }! }{ { n }^{ n } } } converge \forall |\alpha |<e and if so, how can I prove it? Homework Equations { e }^{ x }=\sum _{ n=0 }^{ \infty }{ \frac { { x }^{ n } }{ n! } } The Attempt at a Solution...
  46. C

    Finding infinite series formulae

    Homework Statement This isn't my homework problem, but I'd like to know how to prove \sum _{ n=0 }^{ \infty }{ \frac { n }{ { \alpha }^{ n } } } =\quad \frac { \alpha }{ { (\alpha -1) }^{ 2 } } I only know basic convergence tests (including the integral test), and that \forall |\alpha...
  47. S

    Finding the Sum of an Infinite Series with a Given Radius |x|<1

    Homework Statement I need to find the sum of a given infinite series when |x|<1 (which is the radius of this series) Homework Equations \sum_{n=1}^{∞}(-1)^{n+1}\frac{x^{2n+1}}{4n^2-1} The Attempt at a Solution I've tried to do the following: S'(x) =...
  48. Loren Booda

    LaTeX LaTex and solution for an infinite series

    What is the LaTex and infinite sum for 1-2-1/2+3-1/2-4-1/2+5-1/2 . . . Does it converge anyway? I am too old for this to be a school assignment.
  49. K

    Find the equivalent impedance of an infinite series of resistors and capacitors

    Homework Statement Find the equivalent impedance of the infinite series of resistors and capacitors as shown below -R----R----R----R----...R----... ____C____C____C____B_______C -r----r----r----r----...r----... Homework Equations 2.1. Equivalent resistance of resistors in series : R = R1 + R2...
  50. E

    Sum of the infinite series ((-1)^n * (-7)^n)/n

    Homework Statement Find the sum of infinite the series (-1)^n * 7^n/n! for n=1 to infinity Homework Equations e^x = sum (x^n)/n! for n=0 to infinity The Attempt at a Solution I combined the (-1)^n and the 7^n to make the summation ((-7)^n)/n! for n = 1 to infinity then I changed the lower...
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