Infinite Series

An infinite series is an expression formed by adding the terms of an infinite sequence in a specified order. Its ordinary sum is defined as the limit of its finite partial sums, when that limit exists. Infinite series provide a way to obtain finite values from infinitely many contributions, but they may also diverge. Their convergence and manipulation are central subjects of calculus and mathematical analysis. (openstax.org)

Definition and convergence

For a sequence $a_1,a_2,\ldots$ of real numbers or complex numbers, the associated series is written

\sum_{n=1}^{\infty}a_n=a_1+a_2+a_3+\cdots.

Its $N$ th partial sum is

S_N=\sum_{n=1}^{N}a_n.

The series converges to $S$ if $S_N\to S$ as $N\to\infty$ . More precisely, for every $\varepsilon>0$ , there must be an index $N_0$ such that $|S_N-S|<\varepsilon$ whenever $N\ge N_0$ . Otherwise, the series diverges. Thus, an infinite sum is a limiting process, not a finite addition with an infinitely distant last term. (openstax.org)

A series must be distinguished from its sequence of terms. Convergence of $\sum a_n$ implies $a_n\to0$ , because $a_n=S_n-S_{n-1}$ , but the converse is false: terms can tend to zero while their accumulated sum grows without bound. Divergence can also occur through oscillation rather than unbounded growth. (openstax.org)

The Cauchy criterion expresses convergence without requiring the sum to be known in advance. A real or complex series converges exactly when, for every $\varepsilon>0$ , all sufficiently late finite tails satisfy

\left|\sum_{n=p}^{q}a_n\right|<\varepsilon \qquad(q\ge p).

Equivalently, its partial sums form a Cauchy sequence. This criterion underlies the rigorous treatment of series through the smallness of their tails. (mathshistory.st-andrews.ac.uk)

Fundamental examples

Geometric series

A geometric series has a constant ratio between consecutive terms:

\sum_{n=0}^{\infty}ar^n.

For $r\ne1$ , its partial sum through $n=N$ is

a\frac{1-r^{N+1}}{1-r}.

Consequently,

\sum_{n=0}^{\infty}ar^n=\frac{a}{1-r} \qquad\text{when }|r|<1.

For $a\ne0$ , it diverges when $|r|\ge1$ . In particular,

1+\frac12+\frac14+\frac18+\cdots=2.

The zero series, obtained when $a=0$ , is a trivial exception to the divergence statement. (openstax.org)

Harmonic and $p$ -series

The harmonic series

\sum_{n=1}^{\infty}\frac1n

diverges even though $1/n\to0$ . More generally, for real $p$ , a $p$ -series

\sum_{n=1}^{\infty}\frac1{n^p}

converges exactly when $p>1$ . These series are standard benchmarks for comparison tests. (openstax.org)

For a complex parameter $s$ , the related expression

\zeta(s)=\sum_{n=1}^{\infty}n^{-s}

defines the Riemann zeta function in the region $\operatorname{Re}(s)>1$ . Definitions of that function outside this region require an extension beyond this convergent series representation. (dlmf.nist.gov)

Telescoping series

In a telescoping series, cancellation makes the partial sums especially simple. For example,

\sum_{n=1}^{N}\frac1{n(n+1)} = \sum_{n=1}^{N}\left(\frac1n-\frac1{n+1}\right) = 1-\frac1{N+1}.

Taking the limit gives

\sum_{n=1}^{\infty}\frac1{n(n+1)}=1.

The cancellation is performed first in finite partial sums, avoiding any assumption that infinite expressions may be manipulated like finite ones. (openstax.org)

Absolute and conditional convergence

A series is absolutely convergent if

\sum_{n=1}^{\infty}|a_n|<\infty.

Absolute convergence implies ordinary convergence. A series is conditionally convergent if it converges but its series of absolute values diverges. These distinctions are collected under absolute and conditional convergence. (openstax.org)

The alternating harmonic series is a standard conditionally convergent example:

1-\frac12+\frac13-\frac14+\cdots=\ln2.

Its signs provide cancellation sufficient for convergence, although the corresponding harmonic series of positive terms diverges. (openstax.org)

Absolute convergence is particularly important when changing the order of terms: every rearrangement of an absolutely convergent real or complex series has the same sum. Conditional convergence does not provide this protection. The Riemann rearrangement theorem states that a conditionally convergent real series can be rearranged to converge to any prescribed real number, or to diverge. The order of summation is therefore part of the meaning of a conditionally convergent series. (openstax.org)

Tests for convergence

Most series do not have a convenient formula for their partial sums. Convergence tests establish convergence or divergence without necessarily determining the sum.

Term test: if $a_n$ does not tend to zero, then $\sum a_n$ diverges. If $a_n\to0$ , this test is inconclusive. (openstax.org)
Comparison test: if $0\le a_n\le b_n$ eventually and $\sum b_n$ converges, then $\sum a_n$ converges. If $a_n\ge b_n\ge0$ eventually and $\sum b_n$ diverges, then $\sum a_n$ diverges. (openstax.org)
Limit comparison test: for eventually positive terms, if $a_n/b_n\to L$ with $0<L<\infty$ , the two series either both converge or both diverge. (openstax.org)
Integral test: if $a_n=f(n)$ , where $f$ is eventually positive, continuous, and decreasing, then $\sum a_n$ and the corresponding improper integral have the same convergence behavior. (openstax.org)

The ratio test, assuming the terms are eventually nonzero, uses

L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|.

If $L<1$ , the series converges absolutely; if $L>1$ , it diverges. When $L=1$ , the test is inconclusive. The root test similarly examines $\lim_{n\to\infty}|a_n|^{1/n}$ , when this limit exists, with the same conclusions. These tests detect geometric-type decay or growth. (openstax.org)

For an alternating series

\sum_{n=1}^{\infty}(-1)^{n-1}b_n,

the alternating-series test guarantees convergence if $b_n\ge0$ , $b_n$ is eventually nonincreasing, and $b_n\to0$ . This establishes convergence, not necessarily absolute convergence. (openstax.org)

Series of functions

When the terms are functions, a series takes the form $\sum f_n(x)$ . Pointwise convergence means that the numerical series converges separately at each point. Uniform convergence requires the partial sums to approach their limiting function with an error bound that works simultaneously throughout the specified domain. The Weierstrass $M$ -test guarantees uniform convergence if $|f_n(x)|\le M_n$ everywhere and $\sum M_n$ converges. (dlmf.nist.gov)

A power series has the form

\sum_{n=0}^{\infty}c_n(x-a)^n.

It has a radius of convergence $R$ , possibly zero or infinite: it converges absolutely for $|x-a|<R$ and diverges for $|x-a|>R$ . Boundary points require separate analysis. (openstax.org)

A Taylor series uses coefficients determined by the derivatives of a function:

\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n.

At $a=0$ , it is called a Maclaurin series. For example, the exponential function has the representation

e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}.

Having derivatives of every order does not by itself guarantee that a function equals its Taylor series; equality requires the Taylor remainder to tend to zero. (openstax.org)

Fourier series instead expand functions using sinusoidal or complex exponential terms. Their convergence can depend on the point and on the type of convergence considered. Near jump discontinuities, partial sums can exhibit persistent overshoot, known as the Gibbs phenomenon. (dlmf.nist.gov)

Approximation and error bounds

For a convergent series with sum $S$ , truncation after $N$ terms gives an approximation $S_N$ , with remainder

R_N=S-S_N.

Convergence ensures $R_N\to0$ , but a useful computation also needs information about its size. Under the alternating-series test hypotheses,

|R_N|\le b_{N+1}.

Thus the first omitted term bounds the truncation error. (openstax.org)

For a positive decreasing function satisfying $a_n=f(n)$ , the integral-test remainder estimate is

\int_{N+1}^{\infty}f(x)\,dx \le R_N \le \int_N^{\infty}f(x)\,dx.

For example, applying these bounds to $\sum n^{-2}$ places its remainder between $1/(N+1)$ and $1/N$ . This illustrates why convergence alone does not imply rapid numerical approximation. (openstax.org)

In numerical analysis, convergence acceleration transforms partial sums into a new sequence intended to approach the same limit faster. A limit-preserving transformation must retain the original limit whenever the stipulated convergence conditions hold. (visiblecement.nist.gov)

Divergent series and generalized summation

A divergent series has no ordinary finite sum, but a specified summability method may assign it a generalized value. Such methods must be distinguished from convergence of partial sums. A method is called regular if it agrees with ordinary summation for every convergent series to which it applies. (dlmf.nist.gov)

For example, Cesàro summation considers the arithmetic means of partial sums. For

1-1+1-1+\cdots,

the partial sums alternate between $1$ and $0$ , so the series diverges ordinarily. Their arithmetic means tend to $1/2$ , giving a Cesàro value of $1/2$ . This does not make the original partial sums convergent. (dlmf.nist.gov)

An asymptotic expansion is another distinct use of series notation. Its purpose is to describe approximation as a parameter approaches a specified limit; an infinite expansion need not converge when that parameter is fixed. Finite truncations can nevertheless provide useful approximations. (dlmf.nist.gov)

Historical development

A major development in the theory was the systematic separation of convergent series from formal manipulations of divergent expressions. In his Cours d’analyse of 1821, Augustin-Louis Cauchy gave definitions of convergence and absolute convergence and developed criteria based on limits and the behavior of tails. His work helped establish the rigorous framework in which the validity of an infinite operation must be proved rather than inferred from corresponding finite algebraic operations. (mathshistory.st-andrews.ac.uk)

References

2 Infinite Series — Calculus Volume 2openstax.org
Ch. 5 Introduction — Calculus Volume 2openstax.org
Ch. 5 Key Concepts — Calculus Volume 2openstax.org
Ch. 5 Key Terms — Calculus Volume 2openstax.org
3 The Divergence and Integral Tests — Calculus Volume 2openstax.org
4 Comparison Tests — Calculus Volume 2openstax.org
5 Alternating Series — Calculus Volume 2openstax.org
6 Ratio and Root Tests — Calculus Volume 2openstax.org
1 Power Series and Functions — Calculus Volume 2openstax.org
3 Taylor and Maclaurin Series — Calculus Volume 2openstax.org
DLMF §1.9 Calculus of a Complex Variabledlmf.nist.gov
DLMF §25.2 Definition and Expansionsdlmf.nist.gov