1. ## Series converge?

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?
$$\sum\limits_{n = 1}^{+\infty }{\sin\left({\pi\sqrt{{n^2}+{k^2}}}\right)}$$

2. 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?
$$\sum\limits_{n = 1}^{+\infty }{\sin\left({\pi\sqrt{{n^2}+{k^2}}}\right)}$$

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

3. 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?
$$\sum\limits_{n = 1}^{+\infty }{\sin\left({\pi\sqrt{{n^2}+{k^2}}}\right)}$$

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

$\chi$ $\sigma$