Monday, 3 December 2012

limits - Is there a standard way to compute limlimitsntoinfty(fracn!nn)1/n?



I'm computing the radii of convergence for some complex power series. For one I need to compute
lim




I know the answer is \frac{1}{e}, so the radius is e. But how could you compute this by hand? I tried taking the logarithms and raising e by this logarithm, but it didn't lead me to the correct limit. (This is just practice, not homework.)


Answer



There are two formulas to compute radius of convergence of the series \sum\limits_{n=1}^\infty{c_n}z^n
\frac{1}{R}=\lim\limits_{n\to\infty}|c_n|^{1/n}=\lim\limits_{n\to\infty}\left|\frac{c_{n+1}}{c_n}\right|.
Use the second one.


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