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

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- Feb 8th 2010, 07:59 PMnzm88prove this
if An>=0 and summation(An) converge, then summation ((An)^2) also converge.please...

- Feb 8th 2010, 08:12 PMDrexel28
This is a corollary of a more comprehensive theorem. If $\displaystyle \sum_{n\in\mathbb{N}}a_n$ converges absolutely then $\displaystyle \sum_{n\in\mathbb{N}}a_n^2$ converges as well.

To see this merely note that since $\displaystyle a_n\to0$ there exists some $\displaystyle N\in\mathbb{N}$ such that $\displaystyle N\leqslant n\implies |a_n|<1\implies a_n^2<|a_n|$ and so $\displaystyle \sum_{j=N}^{\infty}a_n^2<\sum_{j=N}^{\infty}|a_n|$ and the conclusion follows. - Feb 8th 2010, 08:12 PMBruno J.
Well from some point on, say for $\displaystyle n>n_0$, we will have $\displaystyle |A_n|<1$. But then $\displaystyle A_n^2<|A_n|$; so just compare the tails of the two series.

- Feb 8th 2010, 08:13 PMBruno J.
Great minds think alike (Nerd)

- Feb 8th 2010, 08:14 PMDrexel28
- Feb 8th 2010, 08:20 PMBruno J.
It'd be an honour to wing-man you! (Rock)