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Math Help - expansion of e^x in Taylor series

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    expansion of e^x in Taylor series

    Am I right saying that the expansion of  e^x in Taylor series is convergent?

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    Quote Originally Posted by rayman View Post
    Am I right saying that the expansion of  e^x in Taylor series is convergent?

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    Yes, you are.
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    thank you
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    More correctly, it is convergent for all x.

    Each term in the series, about x_0, is a_n= \frac{e^{x_0}}{n!}(x- x_0)^n

    Using the ratio test, \left|\frac{a_{n+1}}{a_n}\right|= \frac{e^{x_0}}{(n+1)!}\left|(x- x_0)^{n+1}\right|\frac{n!}{e^{x_0}\left|(x- x_0)^n\right|} = \frac{1}{n+1}\left|x- x_0\right|
    which goes to 0< 1 for all x and all x_0.
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