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Examples of numerical series

In general, a geometric series with initial term a {\displaystyle a} and common ratio r {\displaystyle r},

∑ n = 0 ∞ a r n, {\displaystyle \sum _{n=0}^{\infty }ar^{n},}

converges if and only if | r | < 1 {\textstyle |r|<1}, in which case it converges to a 1 − r {\textstyle {a \over 1-r}}.

The harmonic series is divergent.

the alternating harmonic series, and

− 1 + 1 3 − 1 5 + 1 7 − 1 9 + ⋯ = ∑ n = 1 ∞ (− 1) n 2 n − 1 = − π 4, {\displaystyle -1+{\frac {1}{3}}-{\frac {1}{5}}+{\frac {1}{7}}-{\frac {1}{9}}+\cdots =\sum _{n=1}^{\infty }{\frac {\left(-1\right)^{n}}{2n-1}}=-{\frac {\pi }{4}},}

the Leibniz formula for π. {\displaystyle \pi.}

converges if the sequence b converges to a limit L —as n goes to infinity. The value of the series is then b − L.

converges for p > 1 and diverges for p ≤ 1, which can be shown with the integral criterion described below in convergence tests. As a function of p, the sum of this series is Riemann's zeta function.

and their generalizations (such as basic hypergeometric series and elliptic hypergeometric series) frequently appear in integrable systems and mathematical physics.