# assuming Cauchy product of $\sum_{n=0}^\infty{(-1)^n}$ and $\sum_{n=0}^\infty{(-1)^na_n}$ is convergent find $\sum_{n=0}^\infty{a_n}$

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: $\sum_{n=0}^\infty{a_n}$.

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## 1 Answer

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 $\lim_{n\to \infty}|c_{n}|=0$

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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 at 21:32
Haha it's no problem! –  Daniel Littlewood Jan 16 at 21:34