Counterexample

Is this true?

$\sum{}a_{n}$ converges $\Rightarrow \sum{}a_{n}^{2}$ converges

No — it is false.

The counterexample

$a_{n} = (-1)^{n}/\sqrt{n}$

Why it fails

Alternating series converges; squares give the harmonic series.

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Hypothesis dropped

A theorem applied without checking one of its preconditions.

See real exam options built on this mistake Drill statements like this

Go deeper· Explained in full in Series tests — decision tree and the trap list· Asked in CSIR NET December 2025· From The Real Line › Series: comparison, ratio, root, Raabe, condensation, alternating, rearrangements

Published 23 Aug 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.

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