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Math Help - Power Series

  1. #1
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    Power Series

    Here's the problem:

    \sum_{n=1}^{\infty}\frac{(-1)^n4^nx^n}{\sqrt{n}+10}

    so I use ratio test (obviously):

    \lim_{n\rightarrow \infty}\left | \frac{(-1)^{n+1}4^{n+1}x^{n+1}}{\sqrt{n+1}+10}\times \frac{\sqrt{n}+10}{(-1)^n4^nx^n}\right |

    after some cancellations I have:

    \lim_{n\rightarrow \infty}\left | \frac{-4x(\sqrt{n}+10)}{\sqrt{n+1}+10}\right |

    It seems that I could say that \sqrt{n} cancels, but then again, n + 1 will increase faster than n.

    I can't figure out where to go from here though. Help? Thanks!
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  2. #2
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    Re: Power Series

    Hello, jjtjp!

    Here's the problem:

    \sum_{n=1}^{\infty}\frac{(-1)^n4^nx^n}{\sqrt{n}+10}

    So I use ratio test (obviously):

    \lim_{n\rightarrow \infty}\left | \frac{4^{n+1}x^{n+1}}{\sqrt{n+1}+10}\times \frac{\sqrt{n}+10}{4^nx^n}\right | . We are taking absolute values; drop the (-1)^n.

    After some cancellations I have:

    \lim_{n\rightarrow \infty}\left | \frac{4x(\sqrt{n}+10)}{\sqrt{n+1}+10}\right |

    It seems that I could say that \sqrt{n} cancels,
    but then again, n + 1 will increase faster than n. . This is not true.

    I can't figure out where to go from here though. .Thanks!

    We have: . 4x\cdot\frac{\sqrt{n}+10}{\sqrt{n+1} + 10}


    Divide numerator and denominator by \sqrt{n}:

    . . 4x\cdot\frac{\frac{\sqrt{n}}{\sqrt{n}} + \frac{10}{\sqrt{n}}}{\frac{\sqrt{n+1}}{\sqrt{n}} + \frac{10}{\sqrt{n}}} \;=\;4x\cdot\frac{\sqrt{\frac{n}{n}} + \frac{10}\sqrt{n}}{\sqrt{\frac{n+1}{n}} + \frac{10}{\sqrt{n}}} \;=\;4x\cdot\frac{1 + \frac{10}{\sqrt{n}}}{\sqrt{1\!+\!\frac{1}{n}} + \frac{10}{\sqrt{n}}}

    Therefore: . \lim_{n\to\infty} \left[4x\cdot\frac{1 + \frac{10}{\sqrt{n}}}{\sqrt{1\!+\!\frac{1}{n}} + \frac{10}{\sqrt{n}}}\right] \;=\;4x\cdot\frac{1+0}{\sqrt{1\!+\!0} + 0} \;=\;4x
    Thanks from jjtjp
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  3. #3
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    Re: Power Series

    Thanks so much! I wish I had paid more attention in high school, seems like the trivial stuff always trips me up.
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