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lopus

  • 2 years ago

power serie

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  1. lopus
    • 2 years ago
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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]

  2. satellite73
    • 2 years ago
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    hmm you are expanding around 1, so disregard answer a and e

  3. satellite73
    • 2 years ago
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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 \(x=2\)

  4. satellite73
    • 2 years ago
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    do you know how to do that?

  5. lopus
    • 2 years ago
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    no, i don't

  6. satellite73
    • 2 years ago
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    lets replace \(x\) by 2

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

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

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

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