Series converge?

Hints would be greatly appreciated.

Does the series converge or diverge?
http://latex.codecogs.com/png.latex?...^3}(x)dx}}%5C]

For what k does the series converge?
http://latex.codecogs.com/png.latex?...%5Cright)}%5C]

Thanks in advance!

Quote:

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Originally Posted by Pranas

Hints would be greatly appreciated.

Does the series converge or diverge?
http://latex.codecogs.com/png.latex?...^3}(x)dx}}%5C]

For what k does the series converge?
http://latex.codecogs.com/png.latex?...%5Cright)}%5C]

Thanks in advance!

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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

Quote:

---

Originally Posted by Pranas

Hints would be greatly appreciated.

Does the series converge or diverge?
http://latex.codecogs.com/png.latex?...^3}(x)dx}}%5C]

For what k does the series converge?
http://latex.codecogs.com/png.latex?...%5Cright)}%5C]

Thanks in advance!

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With a little effort You find that is...

http://quicklatex.com/cache3/ql_2b64...705fe30_l3.png (1)

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

http://quicklatex.com/cache3/ql_a974...c490008_l3.png (2)

http://quicklatex.com/cache3/ql_075f...898a78f_l3.png (3)

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

http://quicklatex.com/cache3/ql_9a41...349e505_l3.png (4)

... and obtain...

http://quicklatex.com/cache3/ql_a245...a156294_l3.png

http://quicklatex.com/cache3/ql_136b...9e5af53_l3.png (5)

It is well known that for k>0 http://quicklatex.com/cache3/ql_929d...ea1d1a2_l3.png so that the series (2) converges. The test for the series (3) is left to You...

Kind regards