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Thread: Series Convergence

  1. #1
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    Series Convergence





    What are the values of k that make the series convergent?
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    Re: Series Convergence

    Just a thought, give this a try: Let $\displaystyle u = 2 + e^{2x}$. Then we have that $\displaystyle e^x = \sqrt{u - 2}$.

    Then your sum becomes
    $\displaystyle \sum_{k = 1}^{\infty} \dfrac{\sqrt{u - 2}}{u^k}$

    which looks a lot simpler.

    -Dan
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    Re: Series Convergence

    Quote Originally Posted by topsquark View Post
    Just a thought, give this a try: Let $\displaystyle u = 2 + e^{2x}$. Then we have that $\displaystyle e^x = \sqrt{u - 2}$.

    Then your sum becomes
    $\displaystyle \sum_{k = 1}^{\infty} \dfrac{\sqrt{u - 2}}{u^k}$

    which looks a lot simpler.

    -Dan

    I want x to go from 1 to infinite, not k. I want to figure what are the values of k.
    For example, when k = 0, the series diverges, but k = 1, it converges.
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    Re: Series Convergence

    by plugging random values of k, I observed that the sum increases very very slightly like it will reach finite value for k > 0.5.

    If k <= 0.5, the sum will increase gradually to infinite.

    Does that mean the series is converges when k > 0.5?
    How to prove that mathematically?
    Last edited by joshuaa; May 20th 2019 at 06:51 PM.
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    Re: Series Convergence

    $\displaystyle 0<\frac{e^x}{\left(2+e^{2x}\right)^k}<\frac{e^x}{e ^{2k x}}=\left(e^{1-2k}\right)^x$
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    Re: Series Convergence

    thanks Idea and topsquark.
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