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lopus Group Title

power serie

  • one year ago
  • one year ago

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  1. lopus Group Title
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    \[\sum_{n=1}^{\infty} (-1)^{n+1}*\frac{ (x-1)^n }{ n }\] convergence interval a. (-1,1) b.[0,2) c. (0,2] d.[0,2] e.[-1,1]

    • one year ago
  2. satellite73 Group Title
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    hmm you are expanding around 1, so disregard answer a and e

    • one year ago
  3. satellite73 Group Title
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    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 \(x=2\)

    • one year ago
  4. satellite73 Group Title
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    do you know how to do that?

    • one year ago
  5. lopus Group Title
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    no, i don't

    • one year ago
  6. satellite73 Group Title
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    lets replace \(x\) by 2

    • one year ago
  7. satellite73 Group Title
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    the terms will look like \[(-1)^{n+1}\frac{(2-1)^n}{n}=(-1)^{n+1}\frac{1}{n}\]

    • one year ago
  8. satellite73 Group Title
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    this is an alternating series, whose terms go to zero, and so when you sum it, it will converge

    • one year ago
  9. satellite73 Group Title
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    now we repeat the process with \(x=0\) but before we do, is what i wrote above clear?

    • one year ago
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