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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- January 16th 2011, 10:20 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 . - January 16th 2011, 10:41 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 ;) - January 16th 2011, 10:59 PMProve It
Here's another...

.

Substitute so that . When and when and the integral becomes

provided

.

The rest should follow. - January 16th 2011, 11:07 PMTheCoffeeMachineQuote:

provided

- January 16th 2011, 11:33 PMFernandoRevilla
__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 - January 16th 2011, 11:52 PMProve It
- January 17th 2011, 08:43 AMchiph588@
- January 17th 2011, 09:28 AMFernandoRevilla
It is correct.

: Express__Hint__

Fernando Revilla