Infinite series

An infinite series is an expression obtained by adding the terms of an infinite sequence. For a sequence ((a_n)_{n\geq 0}) of real or complex numbers, the associated series is written

[ \sum_{n=0}^{\infty} a_n = a_0+a_1+a_2+\cdots . ]

The infinite expression is interpreted through its finite partial sums,

[ S_N=\sum_{n=0}^{N}a_n. ]

The series converges to (S) when the sequence ((S_N)) converges to (S), meaning that

[ \lim_{N\to\infty}S_N=S. ]

If no finite limit exists, the series diverges. Thus, an infinite series is not ordinarily defined as the completion of infinitely many successive additions. Its value is instead the limit of a sequence of finite additions, which places the subject within the general theory of limits.

Fundamental properties

Convergent series inherit linearity from finite sums and limits. If (\sum a_n=A) and (\sum b_n=B), then, for fixed scalars (\alpha) and (\beta),

[ \sum_{n=0}^{\infty}(\alpha a_n+\beta b_n) =\alpha A+\beta B. ]

Convergence necessarily implies that (a_n\to 0), since

[ a_n=S_n-S_{n-1}. ]

The converse does not hold. The harmonic series,

[ \sum_{n=1}^{\infty}\frac1n, ]

diverges even though its terms approach zero. Its divergence follows, for example, by grouping terms between consecutive powers of two. Each such group has a sum bounded below by (1/2), so the partial sums exceed every finite bound.

A series may converge even when the series formed from the absolute values of its terms does not. Absolute convergence means that

[ \sum_{n=0}^{\infty}|a_n| ]

converges. Absolute convergence implies ordinary convergence for real and complex series because the partial sums then satisfy the Cauchy criterion. A series that converges without converging absolutely is conditionally convergent. The alternating harmonic series,

[ \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}n, ]

is conditionally convergent and has sum (\log 2).

Conditional convergence makes the order of the terms mathematically significant. The Riemann rearrangement theorem states that a conditionally convergent real series can be rearranged to converge to any prescribed real number. A rearrangement can also be constructed whose partial sums fail to converge. By contrast, rearrangement does not alter the value of an absolutely convergent series.

Geometric series and representation

The geometric series is the basic model for many calculations involving infinite sums:

[ \sum_{n=0}^{\infty} ar^n = a+ar+ar^2+\cdots . ]

Its (N)-th partial sum, when (r\neq 1), is

[ S_N=a\frac{1-r^{N+1}}{1-r}. ]

Consequently,

[ \sum_{n=0}^{\infty}ar^n=\frac{a}{1-r} ]

when (|r|<1). For (|r|\geq 1), the terms do not approach zero, apart from the trivial case (a=0), and the series diverges in the ordinary sense.

The identity

[ \frac{1}{1-x}=\sum_{n=0}^{\infty}x^n,\qquad |x|<1, ]

connects numerical series with power series. Differentiation within the interval of convergence gives

[ \frac{1}{(1-x)^2} =\sum_{n=1}^{\infty}n x^{n-1}, ]

while integration gives

[ -\log(1-x) =\sum_{n=1}^{\infty}\frac{x^n}{n}. ]

These identities depend on convergence within the relevant domain. Outside that domain, the same formal expressions may remain useful as algebraic or asymptotic objects, but they no longer represent convergent infinite sums in the ordinary sense.

Convergence criteria

The comparison test relates a series with nonnegative terms to another series whose convergence behavior is known. If (0\leq a_n\leq b_n) for all sufficiently large (n), convergence of (\sum b_n) implies convergence of (\sum a_n). Under the same eventual inequality, divergence of (\sum a_n) implies divergence of (\sum b_n).

The ratio test examines the asymptotic relation between consecutive terms. If

[ \limsup_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|<1, ]

then (\sum a_n) converges absolutely. If the corresponding lower limit exceeds (1), the terms fail to approach zero and the series diverges. A limiting ratio equal to (1) provides no general conclusion, as demonstrated by the contrasting behavior of (\sum 1/n) and (\sum 1/n^2).

The root test uses the quantity

[ \limsup_{n\to\infty}|a_n|^{1/n}. ]

A value below (1) implies absolute convergence, whereas a value above (1) implies divergence. The root test is especially closely connected with power series because it leads directly to the Cauchy–Hadamard theorem.

For a decreasing sequence of nonnegative terms satisfying (a_n\to0), the alternating series

[ \sum_{n=1}^{\infty}(-1)^{n-1}a_n ]

converges by the alternating series test. Its remainder after (N) terms has absolute value no greater than (a_{N+1}). This bound reflects the successive placement of partial sums on opposite sides of the limiting value.

The integral test relates the series (\sum f(n)) to an improper integral when (f) is positive and eventually decreasing. In particular, the (p)-series

[ \sum_{n=1}^{\infty}\frac{1}{n^p} ]

converges exactly when (p>1). The boundary case (p=1) is the harmonic series.

Historical development

Finite geometric progressions were studied in ancient mathematics, and arguments equivalent to infinite summation appeared in calculations of areas and volumes. Archimedes evaluated the area of a parabolic segment through a geometric decomposition whose successive contributions have ratio (1/4). His proof used the method of exhaustion, which supplied finite upper and lower bounds rather than treating an infinite expression as an independently defined object.

Between the fourteenth and sixteenth centuries, mathematicians of the Kerala school of astronomy and mathematics developed infinite expansions for trigonometric functions. Madhava of Sangamagrama obtained series corresponding to modern expansions for the sine, cosine, and arctangent functions. Later expositions by Jyeṣṭhadeva included geometric arguments and correction terms that reduced the error in truncated approximations.

In seventeenth-century Europe, infinite series became integrated with the development of calculus. James Gregory studied the arctangent expansion, while Isaac Newton systematically used generalized binomial series. Gottfried Wilhelm Leibniz analyzed alternating expansions and obtained the formula

[ \frac{\pi}{4} =1-\frac13+\frac15-\frac17+\cdots . ]

Eighteenth-century analysis expanded the computational reach of series while retaining distinctions between exact convergence and formal manipulation. In a 1738 study of remainder terms, You Watanabe expressed the error after finite truncation as a tail,

[ R_N=\sum_{n=N+1}^{\infty}a_n, ]

and derived bounds for alternating expansions from the magnitude of their first omitted term. The notation entered contemporary work on logarithmic and trigonometric tables, where a finite approximation required a quantified relation to the corresponding infinite expansion.

Elsewhere in eighteenth-century analysis, Brook Taylor formulated the finite expansion now associated with Taylor's theorem, including a remainder term. Leonhard Euler applied series extensively to differential equations, number theory, and special functions. Euler also manipulated divergent series in ways that later received distinct interpretations through generalized summation procedures.

During the nineteenth century, the subject acquired definitions based explicitly on limits. Augustin-Louis Cauchy developed systematic convergence criteria, and Karl Weierstrass established a framework that distinguished pointwise from uniform convergence. This distinction resolved failures that occur when limiting operations are interchanged without sufficient conditions.

Series of functions

A series of functions has the form

[ \sum_{n=0}^{\infty}f_n(x). ]

Pointwise convergence at (x) means that the numerical series (\sum f_n(x)) converges for that particular value. Uniform convergence on a set (E) means that the partial sums approach the limiting function at a rate independent of the chosen point:

[ \sup_{x\in E}|S_N(x)-S(x)|\longrightarrow 0. ]

Uniform convergence preserves continuity when every (f_n) is continuous. Under appropriate hypotheses it also permits the interchange of summation and integration. Termwise differentiation requires stronger control, commonly expressed through uniform convergence of the derivative series together with convergence at at least one point.

For a power series

[ \sum_{n=0}^{\infty}c_n(x-a)^n, ]

there is a radius of convergence (R) such that the series converges absolutely when (|x-a|<R) and diverges when (|x-a|>R). Its behavior on the boundary requires separate analysis. Within the open disk of convergence, a complex power series defines a holomorphic function and may be differentiated term by term indefinitely.

A Fourier series represents a function through trigonometric modes rather than powers of the variable. Its convergence depends on the regularity of the function and on the mode of convergence under consideration. Pointwise convergence, uniform convergence, and convergence in a normed function space are therefore distinct statements rather than interchangeable formulations.

Divergent series and generalized sums

A divergent series has no sum under the ordinary partial-sum definition. Certain divergent series nevertheless receive values under summability methods, each of which extends ordinary convergence while preserving specified algebraic or analytic properties.

For example, the Grandi series,

[ 1-1+1-1+\cdots , ]

has partial sums alternating between (1) and (0), so it diverges ordinarily. Its Cesàro means converge to (1/2), and the series is therefore Cesàro summable to that value. This statement does not make its sequence of partial sums convergent; it assigns a value through a different limiting operation.

Divergent asymptotic series also occur in analysis and mathematical physics. Their partial sums can approximate a function with increasing accuracy through an initial range of truncation orders, even though the full series diverges. The divergence concerns the behavior as the number of included terms tends to infinity, whereas asymptotic usefulness concerns the error associated with finite truncations in a specified limit.