An Abel Summation Maximum

★ 8/10algebra sequences Telescoping inequality

Problem

Let $n$ be a given positive integer, and let $a_1, a_2, \dots, a_n$ be real numbers such that for every $m \leqslant n$ ,

\left|\sum_{k=1}^{m} \frac{a_k}{k}\right| \leqslant 1.

Find the maximum possible value of $\left|\displaystyle\sum_{k=1}^{n} a_k\right|$ .

Answer

Solution

Difficulty	8/10
Topics	algebra, sequences, Telescoping, inequality

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