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January 29th, 2011, 06:26 AM   #1
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Divergence and convergence

I'm not sure if I'm heading down the right path here. I'd gladly appreciate some help.











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January 29th, 2011, 10:44 AM   #2
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Re: Divergence and convergence

Well, we could just come out and sum one.


For a geometric series with ratio r (and I will assume WLOG that the first term is 1),
Consider S_n = 1 + r + r^2 + r^3 +...+ r^n

Then (S_n)r = r + r^2 + r^3 + ... + r^(n+1).

Solve for S, and take a limit.
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January 29th, 2011, 05:23 PM   #3
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Quote:
Originally Posted by DLowry
For a geometric series with ratio r (and I will assume WLOG that the first term is 1),
Consider S_n = 1 + r + r^2 + r^3 +...+ r^n

Then (S_n)r =

Solve for S, and take a limit.



How does this 'take a limit' concept work?
What exactly is meant by without any loss of generality?


I still don't get part a.
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January 29th, 2011, 11:35 PM   #4
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Now

and if |r| < 1, then

So

If |r| > 1, and the series does not converge.

, provided that |r| < 1, a Geometric Progression converges to a sum of
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