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Discuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}{\frac  {2^{n}}{(2n)!}}.

Let us use the ratio test. It works well for series whose terms involve exponential functions and factorial functions. We will compare the terms of the series with the terms right next to them, looking at the limit of this.

\lim _{{n\rightarrow \infty }}{\frac  {a_{{n+1}}}{a_{n}}}=\lim _{{n\rightarrow \infty }}\left|{\frac  {{\frac  {2^{{n+1}}}{(2n+2)!}}}{{\frac  {2^{n}}{(2n)!}}}}\right|=\lim _{{n\rightarrow \infty }}\left|{\frac  {2}{(2n+2)(2n+1)}}\right|=0\,

So, since this limit converges to 0, by the ratio test, our series converges absolutely.

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