Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

This is a question from my analysis midterm, I have only managed to prove that a sum does exist.

Let $(a_n)$ be a sequence of non negative elements such that the Cauchy product of $\sum_{n=0}^\infty{(-1)^n}$ and $\sum_{n=0}^\infty{(-1)^na_n}$ is convergent. Find the sum of the following series: $\sum_{n=0}^\infty{a_n}$.

share|cite|improve this question

1 Answer 1

up vote 3 down vote accepted

The Cauchy product of your two series has general term $c_{n}$, where $$c_{n}=\sum_{k=0}^{n} (-1)^{k}(-1)^{n-k}a_{k}=(-1)^{n}\sum_{k=0}^{n}a_{k}$$ So the partial sums of your $a_{n}$ are just $|c_{n}|$, but the product is convergent so $\lim_{n\to \infty}|c_{n}|=0$

share|cite|improve this answer
thanks a lot, don't know how I could miss that, I got to the same point and I took like half a page more just to find the sum exists ;) – Marek Jan 16 '14 at 21:32
Haha it's no problem! – Daniel Littlewood Jan 16 '14 at 21:34

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.