Problem 30

ThePerfectHacker · #1 $Old$ 07-09-2007, 02:34 PM

Show that $\sum_{n=1}^{\infty}nx^{n-1} = \frac{1}{(1-x)^2}$ for $|x|<1$ without differenciating the geometric power series.

red_dog · #2 $Old$ 07-09-2007, 02:55 PM

Let $S_n=\sum_{k=1}^nkx^{k-1}$
We have $S_n=1+2x+3x^2+\ldots +nx^{n-1}$ (1).
Multiplying (1) with $x$ we have
$xS_n=x+2x^2+3x^3+\ldots +(n-1)x^{n-1}+nx^n$ (2)
Substracting (2) from (1) yields
$S_n(1-x)=1+x+x^2+\ldots +x^{n-1}-nx^n=\frac{1-x^n}{1-x}-nx^n\Rightarrow S_n=\frac{1-x^n}{(1-x)^2}-\frac{nx^n}{1-x}$
Then $\sum_{n=1}^{\infty}nx^{n-1}=\lim_{n\to\infty}S_n=\frac{1}{(1-x)^2}$

ThePerfectHacker · #3 $Old$ 07-10-2007, 05:12 PM

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 $\sum_{n=1}^{\infty} nx^{n-1} = \frac{1}{(1-x)^2}$

Can you prove the following?
Let $f(x) = \sum_{n=0}^{\infty} a_nx^n$ with radius of convergence $R>0$

Then, $f$ is differenciable on $(-R,R)$ and furthermore $f'(x) = \sum_{n=1}^{\infty} na_nx^{n-1}$ .

Note: This works even if $x$ is a complex number, but the proofs are completely anagolous.

Rebesques · #4 $Old$ 07-11-2007, 11:36 AM

Quote:

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

That's ok, we just apply the relevant theorem

ThePerfectHacker · #5 $Old$ 07-11-2007, 11:46 AM

Quote:

Originally Posted by Rebesques $View Post$

That's ok, we just apply the relevant theorem

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.