Series converge?

**Pranas** · May 2nd 2011, 11:52 AM

Hints would be greatly appreciated.

Does the series converge or diverge?
$\sum\limits_{n = 1}^{+\infty }{\int\limits_0^{\frac{\pi }{n}}{{{\sin }^3}(x)dx}}$

For what k does the series converge?
$\sum\limits_{n = 1}^{+\infty }{\sin\left({\pi\sqrt{{n^2}+{k^2}}}\right)}$

Thanks in advance!

**tonio** · May 2nd 2011, 06:27 PM

Originally Posted by Pranas

Hints would be greatly appreciated.

Does the series converge or diverge?
$\sum\limits_{n = 1}^{+\infty }{\int\limits_0^{\frac{\pi }{n}}{{{\sin }^3}(x)dx}}$

For what k does the series converge?
$\sum\limits_{n = 1}^{+\infty }{\sin\left({\pi\sqrt{{n^2}+{k^2}}}\right)}$

Thanks in advance!

sin^3(x) = sin(x)(1 - cos^2(x)) = sin(x) - sin(x)co^2(x) , so that the integral equals -cos(x) + (1/2)cos^3(x) between 0 and pi/n...take it from here

Tonio

**chisigma** · May 2nd 2011, 11:18 PM

Originally Posted by Pranas

Hints would be greatly appreciated.

Does the series converge or diverge?
$\sum\limits_{n = 1}^{+\infty }{\int\limits_0^{\frac{\pi }{n}}{{{\sin }^3}(x)dx}}$

For what k does the series converge?
$\sum\limits_{n = 1}^{+\infty }{\sin\left({\pi\sqrt{{n^2}+{k^2}}}\right)}$

Thanks in advance!

With a little effort You find that is...

(1)

... so that You have to test the convergence of the series...

(2)

(3)

For the (2) You can use the expansion...

(4)

... and obtain...

(5)

It is well known that for k>0

so that the series (2) converges. The test for the series (3) is left to You...

Kind regards

$\chi$ $\sigma$

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