Convergence True or False Proof

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April 26th 2010, 04:17 PM
ThreeLeftsAndHome

Convergence True or False Proof

If $a_k \to 0$ as $k \to \infty$ , and $\displaystyle\sum_{k=1}^{\infty} a_k$ converges, then $a_k \downarrow 0$ as $k \to \infty$ .

I think this is false and my example is $(-1)^k \frac{1}{k^2}$ . What you think?
April 26th 2010, 05:23 PM
Krizalid

Quote:

Originally Posted by ThreeLeftsAndHome

If $a_k \to 0$ as $k \to \infty$ , and $\displaystyle\sum_{k=1}^{\infty} a_k$ converges, then $a_k \downarrow 0$ as $k \to \infty$ .

first information is irrelevant since given the convergence of the series, then $a_k\to0$ as $k\to\infty.$

oh, i think is misread the problem, wonder what does mean $a_k\downarrow0.$
April 26th 2010, 05:31 PM
Defunkt

Quote:

Originally Posted by Krizalid

first information is irrelevant since given the convergence of the series, then $a_k\to0$ as $k\to\infty.$

oh, i think is misread the problem, wonder what does mean $a_k\downarrow0.$

I think it means that $a_k$ is monotone decreasing.
April 26th 2010, 05:38 PM
Krizalid

ah well, but it doesn't say if $a_n\ge0,$ so the counterexample is correct.

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