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Discuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}{\frac  {n4^{n}}{n!}}.

It might not be obvious which grows faster, our numerator or our denominator. The numerator is a linear function multiplied by an exponential whereas the denominator of our fraction is a factorial faction. The ratio test will help us determine which grows more quickly.

\lim _{{n\rightarrow \infty }}\left|{\frac  {{\frac  {(n+1)4^{{n+1}}}{(n+1)!}}}{{\frac  {n4^{n}}{n!}}}}\right|=\lim _{{n\rightarrow \infty }}\left|{\frac  {4(n+1)}{n(n+1)}}\right|=0\,

Thus, the terms go to 0 which means the factorial function grows much more quickly than the linear function multiplied by the exponential function. We also have shown, by the ratio test, that the series converges absolutely.

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