Calc6.46

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Discuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}{\frac  {n^{k}a^{n}}{n!}}.

So here we have a power function of any order, k could be any number, 1, 100, 1,000,000, or even higher, and we also have an exponential function where a can also be any number as high as you want. Yet, by the ratio test, we will show that a simple factorial function will eventually grow to be much greater than even the product of these two.

\lim _{{n\rightarrow \infty }}\left|{\frac  {{\frac  {(n+1)^{k}a^{{n+1}}}{(n+1)!}}}{{\frac  {n^{k}a^{n}}{n!}}}}\right|=\lim _{{n\rightarrow \infty }}\left|{\frac  {(n+1)^{k}a}{n^{k}(n+1)}}\right|=0\,

The degree of the denominator is one greater than the degree of the numerator so this limit does go to 0. Thus, by the ratio test, this series converges absolutely. This also gives us the information that even a normal factorial function grows more quickly than the product of any power function even when multiplied by an exponential function.


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