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# Proof of convergence of power series? Tweet

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1. Proof of convergence of power series?

I'm not entirely sure what I should write to get the marks on part b here, could I get some help?

Thanks
2. Re: Proof of convergence of power series?
You should write what it asks you to write: state the values of for which the series converges and diverges! It should be something you know by heart the values of for which the series converges and diverges. For let so that ; then . Apply the alternating series test.
3. Re: Proof of convergence of power series?
(Original post by nuodai)
You should write what it asks you to write: state the values of for which the series converges and diverges! It should be something you know by heart the values of for which the series converges and diverges. For let so that ; then . Apply the alternating series test.
What about proving it though? That's the sticking point, really.
4. Re: Proof of convergence of power series?
(Original post by everything)
What about proving it though? That's the sticking point, really.
You should know that for to converge we need , so clearly and are out.

For , split it into two cases: prove that it converges for directly, and use this result together with the alternating series test to prove that it converges for .
Last edited by nuodai; 10-08-2012 at 11:02.
5. Re: Proof of convergence of power series?
(Original post by nuodai)
You should know that for to converge we need , so clearly and are out.

For , split it into two cases: prove that it converges for directly, and use this result together with the alternating series test to prove that it converges for .
How do you 'directly' prove that it converges for 0<x<1 ? Like, it's all pretty clear as to why and such, but I struggle to prove things..
6. Re: Proof of convergence of power series?
(Original post by everything)
How do you 'directly' prove that it converges for 0<x<1 ? Like, it's all pretty clear as to why and such, but I struggle to prove things..
It's a geometric series, so you (probably) know that

You need to show that this converges as . Can you see what to do now? (In fact, as it happens, you don't even need the alternating series test here since this proof applies whenever .)
7. Re: Proof of convergence of power series?
(Original post by nuodai)
It's a geometric series, so you (probably) know that

You need to show that this converges as . Can you see what to do now? (In fact, as it happens, you don't even need the alternating series test here since this proof applies whenever .)
Coo, that's what I did. Had no idea if that was what they were looking for though [since it doesn't really have anything to do with alternating series]. Thanks brah
8. Re: Proof of convergence of power series?
And I would upvote your posts but apparently I need to upvote other people first. I guess you helped me with my last question on here too. :P