Infinite series

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An infinite series is a sum of infinitely many terms. Such a sum can have a finite value, and if it has, it is said to converge. The fact that infinite series can converge resolves several of Zeno's paradoxes.

The simplest convergent infinite series is perhaps

1 + 1/2 + 1/4 + 1/8 + 1/16 + ... = 2

It is possible to "visualize" its convergence on the real number line. This series is a geometric series and mathematicians usually write it as

   ∞
   ∑  2-n  =  2
  n=0

Formally, if an infinite series

   ∞
   ∑  an
  n=0

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

         N    
   lim   ∑  an
   N→∞  n=0

exists and is equal to S. If this is not the case, we say the series diverges.

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 montone 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) The series ∑ an of real numbers is called alternating if the signs of the an alternate. Such a series converges if the sequences |an| is monotone decreasing and converges towards 0. The converse is in general not true.

Examples

The series

   ∞   1
   ∑  ---
  n=1  nr 

converges if r > 1 and diverges for r < 1, which can be shown with the integral criterium 5) from above. This series gives rise to the Riemann zeta function.

The geometric series

   ∞    
   ∑  zn
  n=0   

converges if and only if |z| < 1.

Absolute convergence

The sum

   ∞
   ∑  an
  n=0

is said to converge absolutely if the series of absolute values

   ∞
   ∑  |an|
  n=0

converges. In this case, the original series, and all reorderings of it, converge, and converge towards the same sum. If a series is not absolutely convergent, then there is always some reordering of the terms so that the reordered series diverges.

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.