Is the following statement true? Cesàro means converge ⇒ the sequence converges

No. aₙ = (−1)ⁿ Partial averages → 0, sequence diverges.

Counterexample

Is this true?

Cesàro means converge ⇒ the sequence converges

No — it is false.

The counterexample

$a_{n} = (-1)^{n}$

Why it fails

Partial averages → 0, sequence diverges.

Keep practicing

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

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