Hello :D

This can look like a stupid question, but I just can't see the trick (or maybe because I'm up since 5:30am, I'm too tired to think correctly...)

Prove :

Thanks :)

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- May 28th 2008, 03:59 AMMoo[SOLVED] Finding a power series
Hello :D

This can look like a stupid question, but I just can't see the trick (or maybe because I'm up since 5:30am, I'm too tired to think correctly...)

Prove :

Thanks :) - May 28th 2008, 04:36 AMPaulRS
By the Binomial Theorem:

Now prove that:

Remember that - May 28th 2008, 04:48 AMPaulRS
Another way: for all naturals n

Thus:

Then: - May 28th 2008, 08:51 AMThePerfectHacker
Another way. ( is unit circle).

Thus,

Geometric series, .

Now use residue theorem and this integral calculates to . - May 28th 2008, 09:22 AMMoo
- June 20th 2008, 08:09 AMPaulRS
let

Note that:

Thus: let we have:

Thus our solution satisfies the differential equation: which is quite easy to solve (we define )