Basic Analysis I — Real Analysis · Problem 2.5.17

Sequences and Series — Jiri Lebl

Mathematics — Real Analysis · abel-theorem

Problem

(Challenging): Prove Abel's theorem: Theorem. Suppose ∑n=1∞xn\sum_{n=1}^{\infty} x_n is a series whose sequence of partial sums is bounded, {λn}n=1∞\{\lambda_n\}_{n=1}^{\infty} is a sequence with lim⁡n→∞λn=0\lim_{n \to \infty} \lambda_n = 0, and ∑n=1∞∣λn+1−λn∣\sum_{n=1}^{\infty} |\lambda_{n+1} - \lambda_n| is convergent. Then ∑n=1∞λnxn\sum_{n=1}^{\infty} \lambda_n x_n is convergent.

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