I have the following problem: Proof if it is convergence or divergence.

I am trying to do it using the so-called " ratio test "

\(\displaystyle

\sum ^{\infty }_{n=1}\dfrac {n^{n}\cdot n!}{\left( 2n\right) !}

\)

\(\displaystyle

a_n=\dfrac {n^{n}\cdot n!}{\left( 2n\right) !}

\)

\(\displaystyle

a_{n+1} = \dfrac {\left( n+1\right) ^{\left( n+1\right) }.\left( n+1\right) !}{\left( 2\left( n+1\right) \right) !}

\)

\(\displaystyle

\lim _{n\rightarrow \infty }\left[ \dfrac {\left( n+1\right) ^{\left( n+1\right) }\left( n+1\right) !}{\left( 2\left( n+1\right) \right) !}\cdot \dfrac {2n!}{n^{n}\cdot n!}\right]

\)

\(\displaystyle

\lim _{n\rightarrow \infty }\left[ \dfrac {\left( n+1\right) ^{n}\left( n+1\right) \cdot \left( n+1\right) n!}{2.n!\left( n+1\right) }\cdot \dfrac {2n!}{n^{n}\cdot n!}\right]

\)

\(\displaystyle

\lim _{n\rightarrow\infty }\left[ \dfrac {\left( n+1\right) ^{n}\cdot \left( n+1\right) }{n^{n}}\right]

\)

Here I am kind of stuck with the calculation of the limit.

One more similar problem where I am stuck in the same place is :

\(\displaystyle

\sum ^{\infty }_{n=1}\dfrac {n^{n}}{2^{n}\cdot n!}

\)

Can someone explain how should i proceed?

I am trying to do it using the so-called " ratio test "

\(\displaystyle

\sum ^{\infty }_{n=1}\dfrac {n^{n}\cdot n!}{\left( 2n\right) !}

\)

\(\displaystyle

a_n=\dfrac {n^{n}\cdot n!}{\left( 2n\right) !}

\)

\(\displaystyle

a_{n+1} = \dfrac {\left( n+1\right) ^{\left( n+1\right) }.\left( n+1\right) !}{\left( 2\left( n+1\right) \right) !}

\)

\(\displaystyle

\lim _{n\rightarrow \infty }\left[ \dfrac {\left( n+1\right) ^{\left( n+1\right) }\left( n+1\right) !}{\left( 2\left( n+1\right) \right) !}\cdot \dfrac {2n!}{n^{n}\cdot n!}\right]

\)

\(\displaystyle

\lim _{n\rightarrow \infty }\left[ \dfrac {\left( n+1\right) ^{n}\left( n+1\right) \cdot \left( n+1\right) n!}{2.n!\left( n+1\right) }\cdot \dfrac {2n!}{n^{n}\cdot n!}\right]

\)

\(\displaystyle

\lim _{n\rightarrow\infty }\left[ \dfrac {\left( n+1\right) ^{n}\cdot \left( n+1\right) }{n^{n}}\right]

\)

Here I am kind of stuck with the calculation of the limit.

One more similar problem where I am stuck in the same place is :

\(\displaystyle

\sum ^{\infty }_{n=1}\dfrac {n^{n}}{2^{n}\cdot n!}

\)

Can someone explain how should i proceed?

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