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Infinite series

In mathematics, an infinite series is a sum of infinitely many terms. Such a sum can have a finite value; if it has, it is said to converge; if it does not, it is said to diverge. The fact that infinite series can converge resolves several of Zeno's paradoxes.

The simplest convergent infinite series is perhaps

<math>1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots</math>
It is possible to "visualize" its convergence on the real number line: we can imagine a line of length 2, with successive segments marked off of lengths 1, 1/2, 14, etc. There is always room to mark the next segment, because the amount of line remaining is always the same as the last segment marked: when we have marked off 1/2, we still have a piece of length 1/2 unmarked, so we can certainly mark the next 1/4. This argument does not prove that the sum is equal to 2, but it does prove that it is at most 2 -- in other words, the series has an upper bound.

This series is a geometric series and mathematicians usually write it as:

<math>\sum_{n=0}^\infty 2^{-n}=2.</math>

Formally, if an infinite series

<math>\sum_{n=0}^\infty a_n</math>
is given with real (or complex) numbers an, we say that the series converges towards S or that its value is S if the limit
<math>\lim_{N\rightarrow\infty}\sum_{n=0}^N a_n</math>
exists and is equal to S. If there is no such number, then the series is said to diverge.

Table of contents

Some types of infinite series

  • A geometric series is one where each successive term is produced by multiplying the previous term by a constant number. Example: 1 + 1/2 + 1/4 + 1/8 + 1/16...
  • The harmonic series is the series 1 + 1/2 + 1/3 + 1/4 + 1/5...
  • An alternating series[?] is a series where terms alternate signs. Example: 1 - 1/2 + 1/3 + 1/4 - 1/5...

Convergence criteria

1) If the series ∑ an converges, then the sequence (an) converges to 0 for n→∞; the converse is in general not true.

2) If all the numbers an are positive and ∑ bn is a convergent series such that anbn for all n, then ∑ an converges as well. Conversely, if all the bn are positive, anbn for all n and ∑ bn diverges, then ∑ an diverges as well.

3) If the an are positive and there exists a constant C < 1 such that an+1/anC, then ∑ an converges.

4) If the an are positive and there exists a constant C < 1 such that (an)1/nC, then ∑ an converges.

5) If f(x) is a positive monotone decreasing function defined on the interval [1, ∞) with f(n) = an for all n, then ∑ an converges if and only if the integral1 f(x) dx exists.

6) A series of the form ∑ (-1)n an (with an ≥ 0) is called alternating. Such a series converges if the sequence an is monotone decreasing and converges towards 0. The converse is in general not true.

Examples

The series

<math>\sum_{n=1}^\infty\frac{1}{n^r}</math>
converges if r > 1 and diverges for r ≤ 1, which can be shown with the integral criterion 5) from above. As a function of r, the sum of this series is Riemann's zeta function.

The geometric series

<math>\sum_{n=0}^\infty z^n</math>
converges if and only if |z| < 1.

The telescoping series[?]

<math>\sum_{n=1}^\infty (b_n-b_{n+1})</math>
converges if the sequence bn converges to a limit L as n goes to infinity. The value of the series is then b1 - L.

Absolute convergence

The sum

<math>\sum_{n=0}^\infty a_n</math>
is said to converge absolutely if the series of absolute values
<math>\sum_{n=0}^\infty \left|a_n\right|</math>
converges. In this case, the original series, and all reorderings of it, converge, and converge towards the same sum.

If a series converges, but not absolutely, then one can always find a reordering of the terms so that the reordered series diverges. Even more: if the an are real and S is any real number, one can find a reordering so that the reordered series converges with limit S (Riemann).

Power series

Several important functions can be represented as Taylor series; these are infinite series involving powers of the independent variable and are also called power series.

Historically, mathematicians such as Leonhard Euler operated liberally with infinite series, even if they were not convergent. When calculus was put on a sound and correct foundation in the nineteenth century, rigorous proofs of the convergence of series were always required. However, the formal operation with non-convergent series has been retained in rings of formal power series which are studied in abstract algebra. Formal power series are also used in combinatorics to describe and study sequences that are otherwise difficult to handle; this is the method of generating functions.

Generalizations

The notion of series can be defined in every abelian topological group; the most commonly encountered case is that of series in a Banach space.



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