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Thread: Convergence in the form p^n/n!

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    Convergence in the form p^n/n!



    Are these sufficient answers? I feel like maybe they are not convincing enough...
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by davismj View Post


    Are these sufficient answers? I feel like maybe they are not convincing enough...
    They don't convince me at all. Can you explain your reasoning?
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    Quote Originally Posted by Drexel28 View Post
    They don't convince me at all. Can you explain your reasoning?
    well, any complex number $\displaystyle z$ can be written $\displaystyle \rho(e^i\theta)$, and $\displaystyle \rho \in R : \rho = |z|$.

    Therefore $\displaystyle \sum\rho^n/n! = e^\rho$.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by davismj View Post
    well, any complex number $\displaystyle z$ can be written $\displaystyle \rho(e^i\theta)$, and $\displaystyle \rho \in R : \rho = |z|$.

    Therefore $\displaystyle \sum\rho^n/n! = e^\rho$.
    Yes, but $\displaystyle \lim\text{ }a_n=0\implies \sum_{n=1}^{\infty}a_n<\infty$?
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    Quote Originally Posted by Drexel28 View Post
    Yes, but $\displaystyle \lim\text{ }a_n=0\implies \sum_{n=1}^{\infty}a_n<\infty$?
    No, although in this case we know that since $\displaystyle \rho$ is real, the sum is actually equal to $\displaystyle e^\rho$. Should I write that down?

    Thank you for your help!
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by davismj View Post
    No, although in this case we know that since $\displaystyle \rho$ is real, the sum is actually equal to $\displaystyle e^\rho$. Should I write that down?

    Thank you for your help!
    That isn't how to do this. Try using the ratio test.
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    Quote Originally Posted by Drexel28 View Post
    That isn't how to do this. Try using the ratio test.
    I see now. I was making things harder than necessary.
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