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.

View More On Wikipedia.org
  1. W

    Finding the sum of an infinite series

    Homework Statement I am supposed to find the value of the infinite series: \sum_{n=0}^{+\infty}\frac{\pi\cos(n)}{5^n} Homework Equations I asked this question before on this forum and micromass told me that I should use cos(n)=((e^i)^n+(e^(-i))^n)/2. That equation worked and I was...
  2. W

    I hope this helps, let me know if you need any further assistance!

    Homework Statement (Sorry, but I haven't mastered using the sigma notation in these forums yet). Find the sum of the following infinite series: (n=0)^(inf) SIGMA ((pi)cos(n))/(5^n). Homework Equations I tried using the formula S=(a1)/(1-r). I know that a=pi, but I can't find "r."...
  3. K

    Riemann Zeta Function and Pi in Infinite Series

    I was playing around with an infinite series recently and I noticed something peculiar, I was hoping somebody could clarify something for me. Suppose we have an infinite series of the form: \sum^_{n = 1}^{\infty} 1/n^\phi where \phi is some even natural number, it appears that it is always...
  4. S

    Calculus (infinite series) - Gravitational Potential Energy Problem

    Homework Statement [PLAIN]http://img818.imageshack.us/img818/817/potentialenergy.jpg Homework Equations Given above. The Attempt at a Solution I've never had physics in my life and am completely baffled by this problem. I'm only in calculus 3 and am just learning infinite...
  5. K

    Calculating the Limit of an Infinite Series

    Homework Statement 2/3!+4/5!+6/7!+...to infinity is equal to?
  6. M

    Strategy for Testing Series, Infinite Series.

    I think I did this right... \sum_{i=1}^{\infty} \frac{n}{e^{n^2}} I tried it with the root test to no avail. So I then tried it with the Ratio Test and I come to this expression... \lim_{n \to \infty} \frac{(n+1)}{e^{2n} e(n)} ...which is an indeterminate form (infinity over infinity)...
  7. J

    Advanced Calculus Infinite Series

    Homework Statement Suppose that {an} is a monotone decreasing sequence of positive numbers. Show that if the series an converges, then the lim(nan)=0. Homework Equations The Attempt at a Solution I started the proof with the fact that since I know the sequence is monotone decreasing and the...
  8. E

    Convergence of an infinite series

    Homework Statement http://img840.imageshack.us/img840/3609/unleddn.png note that by log(n), i really mean NATURAL log of n Homework Equations it's convergent, but I can't figure out which test to useThe Attempt at a Solution there is no term to the nth power, so ratio test is useless; root...
  9. T

    Infinite Series Solution for Simplifying f(m): Tips and Approximations

    I would like to simplify this series as much as possible f(m)=\sum_{n=0}^{\infty}\frac{m^n (2n)!}{(n!)^3} Approximates would also be fine. One can easily notice that (2n!) / (n!)^2 > 2^n hence I figured out that f(m) > \sum_{n=0}^{\infty}\frac{(2m)^n}{n!}=\exp(2m) but this is not the best...
  10. L

    How to evaluate this infinite series?

    Hello, Could anybody help with this series: $\sum_{n=0}^\infty e^n/(e^n+1)^{a-1},\,\, a>2. $ I tried (without success) to adapt the Riemann integral theorem and the laplace transform. For the latest, I will need to find the inverse laplace transform of $e^n/(e^n+1)^(a-1)$, which does...
  11. D

    Expressing an integral as an infinite series

    Homework Statement \int_0^x \frac{1-cos(t)}{t} Homework Equations The Attempt at a Solution I'm lost completely. If I separate it and then try integrating it has 0 for the ln(x) which has to be wrong.
  12. B

    What is the Limit of the Sequence {[(n+3)/(n+1)]^n} as n Approaches Infinity?

    Basically, find the limit of the sequence: {[(n+3)/(n+1)]^n}, from n=1 to infinity Book says it's supposed to be e^2, and indeed the graph shows that... I'm not sure what to do with the top of the fraction. Working with the bottom and dividing by n, I obtain, lim as n approaches...
  13. H

    Are These Infinite Series Convergent or Divergent?

    Homework Statement I have been straining to find convergence or divergence of a few infinite series. I have tried everything I can think of; ratio test, root test, trying to find a good series for comparison, etc. Here are the formulas for the terms: #1 1 ------------- (ln(n))^ln(n)...
  14. A

    What is the sum of this infinite geometric series?

    Homework Statement ∞ ∑ ( 60^(1/(n+3)) − 60^(1/(n+4)) ) n = 0 Homework Equations I believe this is a geometric series so the sum would equal a/(1-r) The Attempt at a Solution I tried to view it as a geometric series but i had trouble finding a ratio, especially what i thought...
  15. T

    Prove an Infinite Series is Irrational

    Is it possible and is there a general method?
  16. A

    Divergence/convergence infinite series

    Homework Statement show \sum 1/(ln k)^n diverges,for any n. the indexing is k = 2,3,... Homework Equations The Attempt at a Solution Because k > ln k, k^n > ln k^n, and 1/k^n < 1/ln k^n so this is just a p-series, which diverges for p =< 1. So now I need to show it diverges for n > 1...
  17. J

    Converging Infinite Series: Solving ln(1-1/k^2) = -ln(2)

    I have to show that ln(1 - 1/k^2)= - ln(2) I took LCM and separated the equation using ln(A/B) = lnA - lnB How should i proceed now?
  18. D

    Comp Sci Fortran - Help with Truncated Infinite Series Calculator

    Homework Statement I have an assignment to write a program for calculating the sine (and various other functions) using the method of truncated infinite series using DO statements. The DO statement is supposed to run until the difference between the current and last iterations are less than...
  19. Z

    Infinite series converging to natural log

    Prove that sum 1/2^(n+1)*n/(n+1), from n=0 to infinity, converges to 1 - log(2), where log stands for the natural logarithm. I know that the Taylor series for log(x) about x=1 is sum (-1)^(n+1)*(x-1)^n/n, but I don't see how these two statements are consistent. Thanks for any pointers!
  20. I

    What is the Taylor's series for cos(pi/3)?

    \sum [(-1)^n * pi ^2n] / [9^n * (2n)!] = ? Thanks for the help.
  21. A

    Proving Convergence of Infinite Series with Changing Signs

    Homework Statement Show the convergence of the series \sum_{n=1}^{inf}(\frac{1}{n}-\frac{1}{n+x}) of real-valued functions on R - \{-1, -2, -3, ...\} .Homework Equations The Attempt at a Solution I first thought of solving this using telescoping series concept but it didn't work out. Also, I...
  22. D

    Finding the Sum of a Geometric Series with Infinite Terms

    Homework Statement Find the sum of the series: (1/(4^n))+ (((-1)^n)/(3^n)) from n=0 to infinity Homework Equations The Attempt at a Solution I'm not overly familiar with series and am not sure how to approach this. A lot of help guides online talk about testing for...
  23. R

    Convergence/Divergence of an infinite series

    Homework Statement Prove the convergence or divergence of the series \Sigma(\frac{n}{2n+3})^{2} using the Direct Comparison Test. Homework Equations If series A converges and every term in series B is less than the corresponding term in series A, then series B converges. If series C...
  24. F

    Sum of weird infinite series help

    Hi all,SUM of Series from n=2 to infinity of: 1 ------------ (2^n) (n-1) This question is driving me bananas... my tools are Telescoping or Geometric series, but neither seem to work: I've tried everything to get this into a geometric series form and then using the a/1-r formula, but...
  25. L

    Convergence of Infinite Series with Variable Terms?

    Homework Statement Sum from 0 to infinity of (2^n + 6^n)/(2^n6^n) Homework Equations No idea. The Attempt at a Solution I am completely dumbstruck on how to do this one. Could someone give me a hint on where to start? Thanks a lot!
  26. G

    Infinite Series: Convergence & Sum of $\frac{1}{n(n+k)}$

    \sum\frac{1}{n(n+k)} from n=1 to infinity find that the series is convergent and find it's sum. Now I'm a bit confused... I can show it's convergent with k=1 and I attempted the same thing with k by breaking this into partial fractions. But I'm given a harmonic series that is divergent...
  27. T

    Calculating the Value of an Infinite Series

    Homework Statement What is the value of: 1 + (\frac{1}{3})^{2} + (\frac{1}{5})^{2} + (\frac{1}{7})^{2} + (\frac{1}{9})^{2} + ...
  28. matqkks

    Exploring the Physical Intrepretation of Infinite Series

    Is there a physical intrepretation of infinite series? Is there a picture that will explain what is meant by infinite series or is there a tangible application of this concept.
  29. M

    Finding a and b in an infinite series limit comparison test

    Finding "a" and "b" in an infinite series limit comparison test Homework Statement \sum_{n=1}^\infty \frac{\sqrt{n+2}}{2n^2+n+1} How do I identify my a_n and my b_n? In this particular problem you need to use the Limit comparison test which is your "a_n" divided by your "b_n". I...
  30. M

    Why Does the Divergence Test Yield Different Results for These Series?

    Homework Statement \sum_{k=2}^\infty \frac{k^2}{k^2-1} \sum_{n=1}^\infty \frac{1+2^n}{3^n} Homework Equations I know that for the first problem i can apply the Divergence test by finding my limit as K goes to infinity. By doing this i get 1 which does not equal zero so i...
  31. L

    1 last infinite series, power series

    Homework Statement suppose a large number of particles are bouncing back and forth between x=0 and x=1, except that at each endpoint some escape. Let r be the fraction of particles reflected, so then you can assume (1-r) is the number of particles that escape at each wall. Suppose particles...
  32. E

    Diverging and converging infinite series

    I looked through some tutorials to find intervals of divergence and tests for divergence... My series: [PLAIN]http://img843.imageshack.us/img843/4193/51453212.jpg a and x are constants... I did the ratio test and i get \rho=1, so i tried to apply the limit test to see if an is zero or does...
  33. L

    Infinite series, power series problem

    Homework Statement In a water purification process, one-nth of the impurity is removed in the first stage. In each succeeding stage, the amount of impurity removed is one-nth of that removed in the preceding stage. Show that if n=2, the water can be made a pure as you like, but if n=3, at...
  34. N

    Exploring Closed Form Solutions for Infinite Series: Tips and Tricks

    Homework Statement I want to find a closed form formula for: x+2x^2+3x^3+4x^4+\ldots I know that this can be written as: \sum_{n=1}^{\infty}nx^n but I would like to have a closed formula for this. The formula for an infinite geometric series is: \sum_{n=0}^{\infty}x^n =...
  35. C

    Infinite Series Test (Ratio Test get 1)

    Homework Statement Test if the infinite series converge or diverge. Homework Equations \sum_{n=1}^{\infty}\frac{4n+3}{n(n+1)(n+2)} The Attempt at a Solution I tried Ratio test: a_{n+1} = \frac{4n+7}{(n+1)(n+2)(n+3)} a_{n} = \frac{4n+3}{n(n+1)(n+2)} \left|\frac{a_{n+1}}{a_{n}}\right| =...
  36. S

    What is the closed form for the summation of cos(nθ) from -N to N?

    I am having a difficult time working with some of these infinite series. I studied them in calc 2, but that was a few years ago. Could someone help me figure out how to find what the following sum converges to: \sum_{n=-N}^N{cos(n \theta)} Shouldn't there be some property...
  37. S

    Why Does Sum of Infinity Equal Infinity?

    I know that \sum_{n=-\infty}^\infty{1} = \infty But I don't understand why. It seems to me that since the constant inside the summation is not dependent upon n it can be moved outside the summation. Then there is nothing to sum. It seems to me that...
  38. U

    Circle Series Reciprocal: Taking the Reciprocal of an Infinite Series

    I understand how this works: \cos x = \frac{1}{0!} - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!} - \frac{x^{10}}{10!} + \ldots But what about this? \frac{1}{\cos x} = \frac{1}{0!} + \frac{x^2}{2!} + \frac{5x^4}{4!} + \frac{61x^6}{6!} + \frac{1385x^8}{8!} +...
  39. T

    Product of convergent infinite series converges?

    Homework Statement Given two convergent infinite series such that \sum a_n -> L and \sum b_n -> M, determine if the product a_n*b_n converges to L*M. Homework Equations The Attempt at a Solution If know that if a_n -> L this means that the sequence of partial sums of a_n = s_n...
  40. J

    Help with infinite series sum{i=0=>inf} (x^i / (i)^2)

    Hi all, I've been messing around with the product of Poisson distributions and was hoping someone could help me work out a closed form solution for the following convergent infinite series (given x > 0): \sum^{\infty}_{i=0} \frac{x^{i}}{i! \times i!} Many thanks in advance, Jacob.
  41. estro

    Improper integrals and Infinite Series

    Supose a_n=f(n) What is the relationship between convergence or divergence of: \sum_{n=1}^\infty a_n and \int_{1}^{\infty}f(x)dx ? Besides the integral test (which only works for special cases).
  42. K

    Proving the Infinite Series: (xlna)^(n-1)/n!

    Homework Statement Given an infinite series that follows the form [(xlna)^(n-1)]/n! n takes on integers from 0 onwards x all real numbers a all positive real numbers Homework Equations Maclaurin series expansion The Attempt at a Solution In which for the e^x series expansion...
  43. S

    Infinite Series ?COnverge or Diverge

    Infinite Series!??COnverge or Diverge Homework Statement 1. ∑(infinity, k=1) 5k^(-3/2) 2. ∑(infinity, k=1) 1/(k+3) Homework Equations converge or diverge The Attempt at a Solution 1. converges p=3/2... 5/(infin)=> 0 2. diverges p=1 I still don't get why 2 diverges? 1 converges...
  44. A

    Calculating the Sum of Infinite Series: 2^k/k! Method

    how would i go about finding the sum of the infinite series 2^k/k!? its not a geometric so i can't use the formulas for that so i really have no clue. any help would be appreciated
  45. R

    Evaluating an Infinite Series (non geometric)

    Homework Statement http://bit.ly/9N9iLZ Evaluate: lim n-> infinity of Sum (from k = 1 to n) of sqrt(k/n) * 1/n Homework Equations taylor series? The Attempt at a Solution the above = lim n->infinity of Sum (from k = 1 to n) of k^1/2 / n^3/2 k approaches n so n^1/2 / n^3/2 ->...
  46. srfriggen

    Does the Root Test Determine Convergence for \(\sum \frac{5^n}{n+1}\)?

    Homework Statement I don't have the problem in front of me but it was something like "converge or diverge"?: \sum 5^n/(n+1) The Attempt at a Solution I would like to know that if I use the root test, would I get lim n-> \infty 5/(n+1)^1/n = 5/(n+1)^0 = 5/1 = 5 ? I suspect...
  47. J

    Combining Infinite Series: Can I Make These Two Series Start at the Same Point?

    Homework Statement Rewrite the given expression as a sum whose generic term involves xn [m=2 to ∞] ∑m(m-1)amxm-2 + [k=1 to ∞] x∑kakxk-1 Homework Equations None in this problem The Attempt at a Solution To make the first part involve only xn, I can use the substitution n=m-2...
  48. O

    How to evaluate an infinite series with a geometric pattern?

    Homework Statement I'm told to evaluate the following to the thousandths place: \infty \Sigma 7*(0.35)^k k=1 Homework Equations We know that an infinite equation can be expressed as: S\infty=(a1)/1-rn The Attempt at a Solution The first term (a1) is 7 and r=.35 so I can...
  49. B

    Sum of Infinite Series: -3^(n-1)/(8^n) with Geometric Form and Scalar Value

    Homework Statement Find the sum of the following series. SUM (n=1 to inf) -3^(n-1)/(8^n) Homework Equations Possibly fit into ar^n format? [b]3. The Attempt at a Solution [/b I feel there is a way that this fits into a geometric form in which case could use a/(1-r) to find...
  50. L

    Does the series (sin(1/n))/sqrt(n) converge or diverge?

    Infinite series sin(1/n)/n ? Homework Statement does the series (sin (1/n)) / sqrt ( n ) converge or diverge? (series from n = 1 to infinity...) Homework Equations The Attempt at a Solution I thought that for this we could do a comparison of sin (1/n) to a finite number...
Back
Top