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$\displaystyle \sum_{n=0}^{\infty}\frac{7^n}{n!}x^n$

I'm still trying to get the hang of these and feel like I've done something wrong here. After applying the ratio test I end up with:

$\left|7x\right|\lim \limits_{n \to \infty}\left|\frac{1}{n+1}\right|$

That limit is $0$, so does this mean my radius of convergence is $\infty$ and my interval of convergence is $(-\infty, \infty)$?

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    $\begingroup$ Yes, that's right. Note that the series is just $e^{7x}$. $\endgroup$ – carmichael561 Jul 16 '16 at 1:03
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    $\begingroup$ indeed it is, you might also recognize that series as the McLaurin expansion of $e^{7x}$ $\endgroup$ – Doug M Jul 16 '16 at 1:03
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Another way to check:the Cauchy-Hadamard theorem simply gives a formula for the radius of convergence: https://en.wikipedia.org/wiki/Cauchy%E2%80%93Hadamard_theorem

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