I'm not sure if this is worthy of the challenge section, but here goes.

Without using Taylor series/generating functions, show .

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- Jan 16th 2011, 11:20 PMchiph588@Simple series
I'm not sure if this is worthy of the challenge section, but here goes.

Without using Taylor series/generating functions, show . - Jan 16th 2011, 11:41 PMProve It
The MacLaurin Series for for .

So .

And finally .

Edit: Oops, didn't read the question fully stating not to use Taylor series hahaha. Ah well, it stays because it's beautiful ;) - Jan 16th 2011, 11:59 PMProve It
Here's another...

.

Substitute so that . When and when and the integral becomes

provided

.

The rest should follow. - Jan 17th 2011, 12:07 AMTheCoffeeMachineQuote:

provided

- Jan 17th 2011, 12:33 AMFernandoRevilla
__Hints__

**(i)**

**(ii)**The following sequence is convergent

**(iii)**Express

**(iv)**Apply (i) to prove that

which is exactly the sum of the series.

Fernando Revilla - Jan 17th 2011, 12:52 AMProve It
- Jan 17th 2011, 09:43 AMchiph588@
- Jan 17th 2011, 10:28 AMFernandoRevilla
It is correct.

: Express__Hint__

Fernando Revilla