Proof of convergence of power series?
Maths and statistics discussion, revision, exam and homework help.
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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.
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Re: Proof of convergence of power series?What about proving it though? That's the sticking point, really.(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.
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Re: Proof of convergence of power series?You should know that for(Original post by everything)
What about proving it though? That's the sticking point, really.
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. -
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Re: Proof of convergence of power series?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..(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 .
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Re: Proof of convergence of power series?It's a geometric series, so you (probably) know that(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..
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 
.)
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Re: Proof of convergence of power series?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(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 .)
