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Help me please with these 2 questions:

1.Does it converge or diverge? : $$ \sum_{n=2}^{\infty }2^{n}\left ( \frac{n}{n+1} \right )^{n^{2}} $$

2.Check out absolute and conditional convergence of: $x>0 $

$$ \sum_{n=1}^{\infty }\sin (n)\sin \frac{x}{n} $$

Thanks a lot!

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    Part 2 seems to be one of the harder homework type questions. Where did you find part 2? Is this homework? If so, has the class considered some previous problems with terms involving $\sin(n)$?2012-03-16
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    Nope, both of them taken from the web2012-03-20

2 Answers 2

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Hint for 1:

For sufficiently large $n$, $(\frac{n}{n+1})^n = (1 - \frac{1}{n+1})^n \le c$ for some $ 0 \lt c \lt \frac{1}{2}$. Why?

Now trying using the above to prove that your series converges.

For part 2, I believe you can use the Dirichlet Test to prove convergence.

To show that the series does not converge absolutely, use $\sin (x/n) \ge x/2n$ for sufficiently large $n$ and use the fact that at least one of $n$, $n+1$ is more than $\frac{1}{2}$ away from the multiple of $\pi$ which is closest to them.

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    Thank you, it's clear now.2012-03-20
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Part 2

As indicated in Aryabatha's answer, convergence of $\sum(\sin n)\sin(x/n)$ follows from Dirichlet's test: $\sin(x/n)$ is eventually decreasing, converges to $0$ and the partial sums $\sum_{k=1}^n\sin k$ are bounded. To show that it does not converge absolutely, use the inequalities $$ \sin x\ge \frac{2\,x}\pi,\quad|\sin n|\ge\sin^2n=\frac{1-\cos(2\,n)}{2}. $$ Then $$ |(\sin n)\sin\Bigl(\frac{x}{n}\Bigr)|\ge\frac{x}{\pi}\Bigl(\frac1n-\frac{\cos(2\,n)}{n}\Bigr). $$ Again by Dirichlet'e test $\sum_{n=1}^\infty\cos(2\,n)/n$ converges. In particular there exists a constant $A>0$ such that $\Bigl|\sum_{k=1}^n\cos(2\,n)/n\Bigr|\le A$ for all $n$. Then $$ \sum_{k=1}^n|(\sin k)\sin\Bigl(\dfrac{x}{k}\Bigr)|\ge\frac{x}{\pi}\Bigl(\sum_{k=1}^n\frac1k-\sum_{k=1}^n\frac{\cos(2\,n)}{n}\Bigr)\ge\frac{x}{\pi}(\log(n+1)-A). $$

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    Can you elaborate a bit more? Especially the last step...2012-03-17
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    @Aryabhata I have edited the answer.2012-03-17