How to show this sequence converges

**paupsers** · October 30th 2009, 03:07 PM

Prove:

The sequence $b_n=1+2x+3x^2+...+nx^{n-1}, |x|<1$ converges to $(1-x)^{-2}$

I don't even know what method to use to show:
1. that this converges, and
2. that it converges to that value.

Any help with this one?

**Bruno J.** · October 30th 2009, 03:21 PM

Hint :differentiate both sides of the identity

$1+x+x^2+...x^n=\frac{x^{n+1}-1}{x-1}$

**paupsers** · October 30th 2009, 03:22 PM

I can't use differentiation. :-(

**redsoxfan325** · October 30th 2009, 03:36 PM

Originally Posted by paupsers

Prove:

The sequence $b_n=1+2x+3x^2+...+nx^{n-1}, |x|<1$ converges to $(1-x)^{-2}$

I don't even know what method to use to show:
1. that this converges, and
2. that it converges to that value.

Any help with this one?

$S=1+2x+3x^2+...+nx^{n-1}+...\implies \frac{S-1}{x}=2+3x+4x^2+...+nx^{x-2}+...$

Now subtract $1+x+x^2+...+x^{n-2}+...=\frac{1}{1-x}$ from both sides:

$\frac{S-1}{x}-\frac{1}{1-x}=1+2x+3x^2+...+(n-1)x^{n-2}+...$

But the right side of the equation now equals $S$ , so solve for it in the equation below.

$\frac{S-1}{x}-\frac{1}{1-x}=S$

Obviously, this method is not as fast as differentiation.

**Plato** · October 30th 2009, 03:43 PM

Look this is simple.
If $\left| x \right| < 1\, \Rightarrow \,f(x) = \sum\limits_{k = 0}^\infty {x^k } = \frac{1}{{1 - x}}$
then $f'(x) = \sum\limits_{k = 0}^\infty {kx^{k - 1} } = \frac{1}{{\left( {1 - x} \right)^2 }}$

**redsoxfan325** · October 30th 2009, 03:45 PM

Originally Posted by Plato

Look this is simple.
If $\left| x \right| < 1\, \Rightarrow \,f(x) = \sum\limits_{k = 0}^\infty {x^k } = \frac{1}{{1 - x}}$
then $f'(x) = \sum\limits_{k = 0}^\infty {kx^{k - 1} } = \frac{1}{{\left( {1 - x} \right)^2 }}$

That works well, but the only problem is:

Originally Posted by paupsers

I can't use differentiation. :-(

**paupsers** · November 2nd 2009, 11:44 AM

What's a good method to show that the limit exists for this sequence? I've tried showing that it's contractive, but no luck there. Tried the ratio test, didn't work for me. Any ideas?

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