Convergent series

In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence $(a_{1},a_{2},a_{3},\ldots )$ defines a series $S$ that is denoted

For the short story collection, see Convergent Series (short story collection).

The $n$ th partial sum $S n$ is the sum of the first $n$ terms of the sequence; that is,

A series is convergent (or converges) if and only if the sequence $(S_{1},S_{2},S_{3},\dots )$ of its partial sums tends to a limit; that means that, when adding one $a_{k}$ after the other in the order given by the indices, one gets partial sums that become closer and closer to a given number. More precisely, a series converges, if and only if there exists a number $\ell$ such that for every arbitrarily small positive number $\varepsilon$ , there is a (sufficiently large) integer $N$ such that for all $n\geq N$ ,

If the series is convergent, the (necessarily unique) number $\ell$ is called the sum of the series.

The same notation

is used for the series, and, if it is convergent, to its sum. This convention is similar to that which is used for addition: $a + b$ denotes the operation of adding $a$ and $b$ as well as the result of this addition, which is called the sum of $a$ and $b$ .

Any series that is not convergent is said to be divergent or to diverge.

The reciprocals of the produce a divergent series (harmonic series):
${1 \over 1}+{1 \over 2}+{1 \over 3}+{1 \over 4}+{1 \over 5}+{1 \over 6}+\cdots \rightarrow \infty .$

positive integers

Alternating the signs of the reciprocals of positive integers produces a convergent series ():
${1 \over 1}-{1 \over 2}+{1 \over 3}-{1 \over 4}+{1 \over 5}-\cdots =\ln(2)$

alternating harmonic series

The reciprocals of produce a divergent series (so the set of primes is "large"; see divergence of the sum of the reciprocals of the primes):
${1 \over 2}+{1 \over 3}+{1 \over 5}+{1 \over 7}+{1 \over 11}+{1 \over 13}+\cdots \rightarrow \infty .$

prime numbers

The reciprocals of produce a convergent series:
${1 \over 1}+{1 \over 3}+{1 \over 6}+{1 \over 10}+{1 \over 15}+{1 \over 21}+\cdots =2.$

triangular numbers

The reciprocals of produce a convergent series (see e):
${\frac {1}{1}}+{\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{6}}+{\frac {1}{24}}+{\frac {1}{120}}+\cdots =e.$

factorials

The reciprocals of produce a convergent series (the Basel problem):
${1 \over 1}+{1 \over 4}+{1 \over 9}+{1 \over 16}+{1 \over 25}+{1 \over 36}+\cdots ={\pi ^{2} \over 6}.$

square numbers

The reciprocals of produce a convergent series (so the set of powers of 2 is "small"):
${1 \over 1}+{1 \over 2}+{1 \over 4}+{1 \over 8}+{1 \over 16}+{1 \over 32}+\cdots =2.$

powers of 2

The reciprocals of produce a convergent series:
${1 \over 1}+{1 \over n}+{1 \over n^{2}}+{1 \over n^{3}}+{1 \over n^{4}}+{1 \over n^{5}}+\cdots ={n \over n-1}.$

powers of any n>1

Alternating the signs of reciprocals of also produces a convergent series:
${1 \over 1}-{1 \over 2}+{1 \over 4}-{1 \over 8}+{1 \over 16}-{1 \over 32}+\cdots ={2 \over 3}.$

powers of 2

Alternating the signs of reciprocals of powers of any n>1 produces a convergent series:
${1 \over 1}-{1 \over n}+{1 \over n^{2}}-{1 \over n^{3}}+{1 \over n^{4}}-{1 \over n^{5}}+\cdots ={n \over n+1}.$

The reciprocals of produce a convergent series (see ψ):
${\frac {1}{1}}+{\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{8}}+\cdots =\psi .$

Convergent series

positive integers

alternating harmonic series

prime numbers

triangular numbers

factorials

square numbers

powers of 2

powers of any n>1

powers of 2

Fibonacci numbers

Normal convergence

List of mathematical series

"Series"

Riemann Series Theorem