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Abel's Inequality

Let $\{f_n\}$ and $\{a_n\}$ be Sequences with $f_n \geq f_{n+1} > 0$ for $n=1$, 2, ..., then

\begin{displaymath}
\left\vert{\sum_{n=1}^m a_n f_n}\right\vert \leq A f_1,
\end{displaymath}

where

\begin{displaymath}
A = \max\{\vert a_1\vert,\left\vert{a_1+a_2}\right\vert,\ldots,\left\vert{a_1+a_2+\ldots+a_m}\right\vert\}.
\end{displaymath}




© 1996-9 Eric W. Weisstein
1999-05-25