Exercise 1474Calculus › Single-variable calculus › Series › Convergence testsL1Added 2 Nov 2026

A Cauchy product that diverges

#Cauchy product#conditional convergence#Mertens theorem

Statement

Let $a_n=\dfrac{(-1)^n}{\sqrt{n+1}}$ for $n\ge0$ . The series $\sum a_n$ converges (alternating series test). Show that its Cauchy product with itself,
$\sum_{n\ge0}c_n,\qquad c_n=\sum_{k=0}^na_ka_{n-k},$
diverges.

Hint

Show that $|c_n|\ge1$ for all $n$ , so $c_n\not\to0$ .

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Solution

The terms. $a_ka_{n-k}=\dfrac{(-1)^n}{\sqrt{(k+1)(n-k+1)}}$ , so
$c_n=(-1)^n\sum_{k=0}^n\frac1{\sqrt{(k+1)(n-k+1)}}.$
All the terms in the sum have the same sign: no cancellation.

A bound. By AM-GM, $\sqrt{(k+1)(n-k+1)}\le\frac{(k+1)+(n-k+1)}2=\frac{n+2}2$ . Hence
$|c_n|\ge(n+1)\cdot\frac2{n+2}\ge1\quad\text{for all }n\ge0.$

Conclusion. $c_n\not\to0$ , so $\sum c_n$ diverges by the divergence test.

Remark. The product of two convergent series is not always convergent! *Mertens' theorem* says it converges if at least one of the two series converges absolutely; here neither does.

A Cauchy product that diverges

Statement

Solution

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