# Convergence True or False Proof

• Apr 26th 2010, 04:17 PM
ThreeLeftsAndHome
Convergence True or False Proof
If $\displaystyle a_k \to 0$ as $\displaystyle k \to \infty$, and $\displaystyle \displaystyle\sum_{k=1}^{\infty} a_k$ converges, then $\displaystyle a_k \downarrow 0$ as $\displaystyle k \to \infty$.

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

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

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

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

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

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

I think it means that $\displaystyle a_k$ is monotone decreasing.
• Apr 26th 2010, 05:38 PM
Krizalid
ah well, but it doesn't say if $\displaystyle a_n\ge0,$ so the counterexample is correct.