lopus
  • lopus
power serie
Mathematics
  • Stacey Warren - Expert brainly.com
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jamiebookeater
  • jamiebookeater
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lopus
  • lopus
\[\sum_{n=1}^{\infty} (-1)^{n+1}*\frac{ (x-1)^n }{ n }\] convergence interval a. (-1,1) b.[0,2) c. (0,2] d.[0,2] e.[-1,1]
anonymous
  • anonymous
hmm you are expanding around 1, so disregard answer a and e
anonymous
  • anonymous
radius of convergence is 1, so it is one of the middle three your job is to check at the endpoints, and see if it converges at \(x=0\) and if it converges at \(x=2\)

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anonymous
  • anonymous
do you know how to do that?
lopus
  • lopus
no, i don't
anonymous
  • anonymous
lets replace \(x\) by 2
anonymous
  • anonymous
the terms will look like \[(-1)^{n+1}\frac{(2-1)^n}{n}=(-1)^{n+1}\frac{1}{n}\]
anonymous
  • anonymous
this is an alternating series, whose terms go to zero, and so when you sum it, it will converge
anonymous
  • anonymous
now we repeat the process with \(x=0\) but before we do, is what i wrote above clear?

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