# Thread: prove this

1. ## prove this

if An>=0 and summation(An) converge, then summation ((An)^2) also converge.please...

2. Originally Posted by nzm88
if An>=0 and summation(An) converge, then summation ((An)^2) also converge.please...
This is a corollary of a more comprehensive theorem. If $\sum_{n\in\mathbb{N}}a_n$ converges absolutely then $\sum_{n\in\mathbb{N}}a_n^2$ converges as well.

To see this merely note that since $a_n\to0$ there exists some $N\in\mathbb{N}$ such that $N\leqslant n\implies |a_n|<1\implies a_n^2<|a_n|$ and so $\sum_{j=N}^{\infty}a_n^2<\sum_{j=N}^{\infty}|a_n|$ and the conclusion follows.

3. Well from some point on, say for $n>n_0$, we will have $|A_n|<1$. But then $A_n^2<|A_n|$; so just compare the tails of the two series.

4. Great minds think alike

5. Originally Posted by Bruno J.
Great minds think alike
Agreed. We'll probably be in a think-tank someday together. We can then go to bars and try to impress women with our bridling intellect and manish good looks.

6. It'd be an honour to wing-man you!