Wiki Mobile by Verkata

$\sum_{n=1}^\infty \frac{1}{2^n} = \frac{1}{2}+ \frac{1}{4}+ \frac{1}{8}+\cdots+ \frac{1}{2^n}+\cdots.$

The terms of the series are often produced according to a certain rule, such as by a formula, or by an algorithm. As there are an infinite number of terms, this notion is often called an infinite series. Unlike finite summations, infinite series need tools from mathematical analysis to be fully understood and manipulated. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics and computer science.

Basic properties

Series can be composed of terms from any one of many different sets including real numbers, complex numbers, and functions. The definition given here will be for real numbers, but can be generalized.

Given an infinite sequence of real numbers { a_n }, define

$S_N =\sum_{n=0}^N a_n=a_0+a_1+a_2+\cdots+a_N.$

Call S_N the partial sum to N of the sequence { a_n }, or partial sum of the series. A series is the sequence of partial sums, { S_N }.

Potential confusion

When talking about series, one can refer either to the sequence { S_N } of the partial sums, or to the sum of the series,

$\sum_{n=0}^\infty a_n$

i.e., the limit of the sequence of partial sums (see the formal definition in the next section) – it is clear which one is meant from context. To make a distinction between these two completely different objects (sequence vs. summed value), one sometimes omits the limits (atop and below the sum's symbol), as in

$\sum_{n} a_n\$

in order to refer to the formal series, that may or may not have a definite sum.

Convergent series

A series ∑a_n is said to 'converge' or to 'be convergent' when the sequence S_N of partial sums has a finite limit. If the limit of S_N is infinite or does not exist, the series is said to diverge. When the limit of partial sums exists, it is called the sum of the series

$\sum_{n=0}^\infty a_n = \lim_{N\to\infty} S_N = \lim_{N\to\infty} \sum_{n=0}^N a_n.$

The easiest way that an infinite series can converge is if all the a_n are zero for n sufficiently large. Such a series can be identified with a finite sum, so it is only infinite in a trivial sense.

Working out the properties of the series that converge even if all the terms are non-zero is the essence of the study of series. Consider the example

$1 + \frac{1}{2}+ \frac{1}{4}+ \frac{1}{8}+\cdots+ \frac{1}{2^n}+\cdots.$

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, ½, ¼, 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 ½, we still have a piece of length ½ unmarked, so we can certainly mark the next ¼. This argument does not prove that the sum is equal to 2 (although it is), but it does prove that it is at most 2. In other words, the series has an upper bound. Proving that the series is equal to 2 requires only elementary algebra, however. If the series is denoted S, it can be seen that

$S/2 = \frac{1+ \frac{1}{2}+ \frac{1}{4}+ \frac{1}{8}+\cdots}{2} = \frac{1}{2}+ \frac{1}{4}+ \frac{1}{8}+ \frac{1}{16} +\cdots$

Therefore,

$S-S/2 = 1 \Rightarrow S = 2.\,\!$

Mathematicians extend the idiom discussed earlier to other, equivalent notions of series. For instance, when we talk about a recurring decimal, as in

$x = 0.111\dots \,$

we are talking, in fact, just about the series

$\sum_{n=1}^\infty \frac{1}{10^n}$

But since these series always converge to real numbers (because of what is called the completeness property of the real numbers), to talk about the series in this way is the same as to talk about the numbers for which they stand. In particular, it should offend no sensibilities if we make no distinction between 0.111… and ¹/₉. Less clear is the argument that 9 × 0.111… = 0.999… = 1, but it is not untenable when we consider that we can formalize the proof knowing only that limit laws preserve the arithmetic operations. See 0.999... for more.

Examples

A geometric series is one where each successive term is produced by multiplying the previous term by a constant number. Example:

$1 + {1 \over 2} + {1 \over 4} + {1 \over 8} + {1 \over 16} + \cdots=\sum_{n=0}^\infty{1 \over 2^n}.$ In general, the geometric series $\sum_{n=0}^\infty z^n$ converges if and only if |z| < 1.

The harmonic series is the series

$1 + {1 \over 2} + {1 \over 3} + {1 \over 4} + {1 \over 5} + \cdots =\sum_{n=1}^\infty {1 \over n}.$ The harmonic series is divergent.

An alternating series is a series where terms alternate signs. Example:

$1 - {1 \over 2} + {1 \over 3} - {1 \over 4} + {1 \over 5} - \cdots =\sum_{n=1}^\infty (-1)^{n+1} {1 \over n}.$

The series

$\sum_{n=1}^\infty\frac{1}{n^r}$ converges if r > 1 and diverges for r ≤ 1, which can be shown with the integral criterion described below in convergence tests. As a function of r, the sum of this series is Riemann's zeta function.

A telescoping series

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

Properties of series

Series are classed not only by whether they converge or diverge: they can also be split up based on the properties of the terms a_n (absolute or conditional convergence); type of convergence of the series (pointwise, uniform); the class of the term a_n (whether it is a real number, arithmetic progression, trigonometric function); etc.

Non-negative terms

When a_n is a non-negative real number for every n, the sequence S_N of partial sums is non-decreasing. It follows that a series ∑a_n with non-negative terms converges if and only if the sequence S_N of partial sums is bounded.

For example, the series

$\sum_{n \ge 1} \frac{1}{n^2}$

is convergent, because the inequality

$\frac1 {n^2} \le \frac{1}{n-1} - \frac{1}{n}, \quad n \ge 2,$

and a telescopic sum argument imply that the partial sums are bounded by 2.

Absolute convergence

Main article: Absolute convergence

A series

$\sum_{n=0}^\infty a_n$

is said to converge absolutely if the series of absolute values

$\sum_{n=0}^\infty \left|a_n\right|$

converges. It can be proved that this is sufficient to make not only the original series converge to a limit, but also for any reordering of it to converge to the same limit.

Conditional convergence

Main article: Conditional convergence

A series of real or complex numbers is said to be conditionally convergent (or semi-convergent) if it is convergent but not absolutely convergent. A famous example is the alternating series

$\sum\limits_{n=1}^\infty {(-1)^{n+1} \over n} = 1 - {1 \over 2} + {1 \over 3} - {1 \over 4} + {1 \over 5} - \cdots$

which is convergent (and its sum is equal to ln 2), but the series formed by taking the absolute value of each term is the divergent harmonic series. The Riemann series theorem says that any conditionally convergent series can be reordered to make a divergent series, and moreover, if the a_n are real and S is any real number, that one can find a reordering so that the reordered series converges with sum equal to S.

Abel's test is an important tool for handling semi-convergent series. If a series has the form

$\sum a_n = \sum \lambda_n b_n$

where the partial sums B_N = b₀ + ··· + b_n are bounded, λ_n has bounded variation, and lim λ_n B_n exists:

$\sup_N \Bigl| \sum_{n=0}^N b_n \Bigr| < \infty, \ \ \sum |\lambda_{n+1} - \lambda_n| < \infty\ \text{and} \ \lambda_n B_n \ \text{converges,}$

then the series ∑ a_n is convergent. This applies to the pointwise convergence of many trigonometric series, as in

$\sum_{n=2}^\infty \frac{\sin(n x)}{\ln n}$

with 0 < x < 2π. Abel's method consists in writing b_n+1 = B_n+1 − B_n, and in performing a transformation similar to integration by parts (called summation by parts), that relates the given series ∑ a_n to the absolutely convergent series

$\sum (\lambda_n - \lambda_{n+1}) \, B_n.$

Convergence tests

Main article: convergence tests

n-th term test: If lim_n→∞ a_n ≠ 0 then the series diverges.
Comparison test 1: If ∑b_n is an absolutely convergent series such that |a_n | ≤ C |b_n | for some number C and for sufficiently large n , then ∑a_n converges absolutely as well. If ∑|b_n | diverges, and |a_n | ≥ |b_n | for all sufficiently large n , then ∑a_n also fails to converge absolutely (though it could still be conditionally convergent, e.g. if the a_n alternate in sign).
Comparison test 2: If ∑b_n is an absolutely convergent series such that |a_n+1 /a_n | ≤ |b_n+1 /b_n | for sufficiently large n , then ∑a_n converges absolutely as well. If ∑|b_n | diverges, and |a_n+1 /a_n | ≥ |b_n+1 /b_n | for all sufficiently large n , then ∑a_n also fails to converge absolutely (though it could still be conditionally convergent, e.g. if the a_n alternate in sign).
Ratio test: If |a_n+1/a_n| approaches a number less than one as n approaches infinity, then ∑a_n converges absolutely. When the limit of the ratio is 1, convergence can sometimes be determined as well.
Root test: If there exists a constant C < 1 such that |a_n|^1/n ≤ C for all sufficiently large n, then ∑a_n converges absolutely.
Integral test: if ƒ(x) is a positive monotone decreasing function defined on the interval [1, ∞) with ƒ(n) = a_n for all n, then ∑a_n converges if and only if the integral ∫₁^∞ ƒ(x) dx is finite.
Cauchy's condensation test: If a_n is non-negative and non-increasing, then the two series ∑a_n and ∑2^ka_(2^k) are of the same nature: both convergent, or both divergent.
Alternating series test: A series of the form ∑(−1)ⁿ a_n (with a_n ≥ 0) is called alternating. Such a series converges if the sequence a_n is monotone decreasing and converges to 0. The converse is in general not true.
For some specific types of series there are more specialized convergence tests, for instance for Fourier series there is the Dini test.

Series of functions

Main article: Function series

A series of real- or complex-valued functions

$\sum_{n=0}^\infty f_n(x)$

converges pointwise on a set E, if the series converges for each x in E as an ordinary series of real or complex numbers. Equivalently, the partial sums

$s_N(x) = \sum_{n=0}^N f_n(x)$

converge to ƒ(x) as N → ∞ for each x ∈ E.

A stronger notion of convergence of a series of functions is called uniform convergence. The series converges uniformly if it converges pointwise to the function ƒ(x), and the error in approximating the limit by the Nth partial sum,

Contents