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Hardy-Littlewood Tauberian Theorem

Let $a_n\geq 0$ and suppose

\begin{displaymath}
\sum_{n=1}^\infty a_ne^{-an}\sim{1\over a}
\end{displaymath}

as $a\to 0^+$. Then

\begin{displaymath}
\sum_{n\leq x} a_n\sim x
\end{displaymath}

as $x\to\infty$.

See also Tauberian Theorem


References

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 118-119, 1994.




© 1996-9 Eric W. Weisstein
1999-05-25