Lambert summation

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In mathematical analysis, Lambert summation is a summability method for a class of divergent series.

Definition[edit]

A series \sum a_n is Lambert summable to A, written \sum a_n = A \,(\mathrm{L}), if

\lim_{r \rightarrow 1-} (1-r) \sum_{n=1}^\infty \frac{n a_n r^n}{1-r^n} = A . \,

If a series is convergent to A then it is Lambert summable to A (an Abelian theorem).

Examples[edit]

  • \sum_{n=1}^\infty \frac{\mu(n)}{n} = 0 (\mathrm{L}), where μ is the Möbius function. Hence if this series converges at all, it converges to zero.

See also[edit]

References[edit]