I'm computing the radii of convergence for some complex power series. For one I need to compute
limn→∞(n!nn)1/n.
I know the answer is 1e, 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 ∞∑n=1cnzn
1R=limn→∞|cn|1/n=limn→∞|cn+1cn|.
Use the second one.
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