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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- May 2nd 2011, 12:52 PMPranasSeries 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! - May 2nd 2011, 07:27 PMtonio
- May 3rd 2011, 12:18 AMchisigma
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