# Convergent series

{{#invoke:Hatnote|hatnote}}Template:Main other In mathematics, a series is the sum of the terms of a sequence of numbers.

$S_{n}=\sum _{k=1}^{n}a_{k}.$ A series is convergent if the sequence of its partial sums $\left\{S_{1},\ S_{2},\ S_{3},\dots \right\}$ converges; in other words, it approaches a given number. In more formal language, a series converges if there exists a limit $\ell$ such that for any arbitrarily small positive number $\varepsilon >0$ , there is a large integer $N$ such that for all $n\geq \ N$ ,

$\left|S_{n}-\ell \right\vert \leq \ \varepsilon .$ Any series that is not convergent is said to be divergent.

## Convergence tests

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There are a number of methods of determining whether a series converges or diverges.

However, if,

$\lim _{n\to \infty }\left|{\frac {a_{n+1}}{a_{n}}}\right|=r.$ If r < 1, then the series converges. If r > 1, then the series diverges. If r = 1, the ratio test is inconclusive, and the series may converge or diverge.

Root test or nth root test. Suppose that the terms of the sequence in question are non-negative. Define r as follows:

$r=\limsup _{n\rightarrow \infty }{\sqrt[{n}]{|a_{n}|}},$ where "lim sup" denotes the limit superior (possibly ∞; if the limit exists it is the same value).

If r < 1, then the series converges. If r > 1, then the series diverges. If r = 1, the root test is inconclusive, and the series may converge or diverge.

The ratio test and the root test are both based on comparison with a geometric series, and as such they work in similar situations. In fact, if the ratio test works (meaning that the limit exists and is not equal to 1) then so does the root test; the converse, however, is not true. The root test is therefore more generally applicable, but as a practical matter the limit is often difficult to compute for commonly seen types of series.

Integral test. The series can be compared to an integral to establish convergence or divergence. Let $f(n)=a_{n}$ be a positive and monotone decreasing function. If

$\int _{1}^{\infty }f(x)\,dx=\lim _{t\to \infty }\int _{1}^{t}f(x)\,dx<\infty ,$ then the series converges. But if the integral diverges, then the series does so as well.

Alternating series test. Also known as the Leibniz criterion, the alternating series test states that for an alternating series of the form $\sum _{n=1}^{\infty }a_{n}(-1)^{n}$ , if $\left\{a_{n}\right\}$ is monotone decreasing, and has a limit of 0 at infinity, then the series converges.

## Conditional and absolute convergence

File:ExpConvergence.gif
Illustration of the absolute convergence of the power series of Exp[z] around 0 evaluated at z = Exp[[[:Template:Frac]]]. The length of the line is finite. Illustration of the conditional convergence of the power series of log(z+1) around 0 evaluated at z = exp((π−Template:Frac)i). The length of the line is infinite.
$\sum _{n=1}^{\infty }a_{n}\leq \ \sum _{n=1}^{\infty }\left|a_{n}\right\vert .$ If the series $\sum _{n=1}^{\infty }\left|a_{n}\right\vert$ converges, then the series $\sum _{n=1}^{\infty }a_{n}$ is absolutely convergent. An absolutely convergent sequence is one in which the length of the line created by joining together all of the increments to the partial sum is finitely long. The power series of the exponential function is absolutely convergent everywhere.

If the series $\sum _{n=1}^{\infty }a_{n}$ converges but the series $\sum _{n=1}^{\infty }\left|a_{n}\right\vert$ diverges, then the series $\sum _{n=1}^{\infty }a_{n}$ is conditionally convergent. The path formed by connecting the partial sums of a conditionally convergent series is infinitely long. The power series of the logarithm is conditionally convergent.

The Riemann series theorem states that if a series converges conditionally, it is possible to rearrange the terms of the series in such a way that the series converges to any value, or even diverges.

## Uniform convergence

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$s_{n}(x)=\sum _{k=1}^{n}f_{k}(x)$ converges uniformly to f.

There is an analogue of the comparison test for infinite series of functions called the Weierstrass M-test.

## Cauchy convergence criterion

The Cauchy convergence criterion states that a series

$\sum _{n=1}^{\infty }a_{n}$ $\left|\sum _{k=m}^{n}a_{k}\right|<\varepsilon ,$ which is equivalent to

$\lim _{n\to \infty \atop m\to \infty }\sum _{k=n}^{n+m}a_{k}=0.$ 