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Math Help - Help solving taylor series.

  1. #1
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    Help solving taylor series.

    I've just started working with Taylor series and can't figure out how to find the series of the following function:

    g(x)= \int_0^x [2e^v+3e^{-v}] dv

    about x=0

    Could anyone please show me the solution?
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  2. #2
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    Quote Originally Posted by goblinf View Post
    I've just started working with Taylor series and can't figure out how to find the series of the following function:

    g(x)= \int_0^x [2e^v+3e^{-v}] dv

    about x=0

    Could anyone please show me the solution?


    \int\limits_0^x [2e^v+3e^{-v}] dv=2e^x-2-3e^{-x}+3=2e^x-3e^{-x}+1 . Now just remember that

    \displaystyle{e^x=\sum\limits^\infty_{k=0}\frac{x^  k}{k!}\,,\,\,\forall x\in\mathbb{R}}

    Tonio
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  3. #3
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    And for which values of x does the series converge?
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  4. #4
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    Quote Originally Posted by goblinf View Post
    And for which values of x does the series converge?
    Tonio stated that after the series, surely? It holds for all x.
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  5. #5
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    Quote Originally Posted by TheCoffeeMachine View Post
    Tonio stated that after the series, surely? It holds for all x.
    Could you please describe this a bit more. How do you see that it holds for all x?
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  6. #6
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    It's well known that the Taylor series for \displaystyle e^x is convergent for all \displaystyle x.

    But this is also relatively easy to show using the ratio test.
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