Show that forwithoutdifferenciating the geometric power series.

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- Jul 9th 2007, 01:34 PMThePerfectHackerProblem 30
Show that for

**without**differenciating the geometric power series. - Jul 9th 2007, 01:55 PMred_dog
Let

We have (1).

Multiplying (1) with we have

(2)

Substracting (2) from (1) yields

Then - Jul 10th 2007, 04:12 PMThePerfectHacker
Since you answered so quickly, here is a second challenge. I am not sure if is a fair question to ask but I post it anyway.

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Having shown that

Can you prove the following?

Let with radius of convergence

Then, is differenciable on and furthermore .

Note: This works even if is a complex number, but the proofs are completely anagolous. - Jul 11th 2007, 10:36 AMRebesquesQuote:

I am not sure if is a fair question to ask but I post it anyway.

- Jul 11th 2007, 10:46 AMThePerfectHacker
I am not sure what you mean. In my Analysis class we proved differenciation term-by-term by working backwards with the Fundamental Theorem of Calculus after proving integration term-by-term. And that was just Real fucntions.

By my Complex Analysis book had a really nice approach to this proof. Without using more advandeced techiqnues. It was able to prove term-by-term differenciation by using the series posted above. It was a long but straightforward derivation. That is what I am asking to show.