test the convergence of series 1 + 2^p/2! + 3^p/3! + 4^p/4! convergence of infinite series | convergence of series question and answer | solved problems on convergence and divergence

convergence question-  test the convergence of series 1 + 2^p/2! + 3^p/3!  + 4^p/4! solved problems on convergence and divergence 

1 + \frac{ 2^p} {2!} + \frac {3^p} {3!} +  \frac {4^p} {4!} +...

convergence test answers

U_n = \frac { n^p } { n ! }\;\; and \; \;  U_{ n+1}= \frac{ (n+1)^p } { (n+1)!}

\frac{U_n}{U_{n+1}}=\frac{n^p}{n!}.\frac{(n+1)!}{(n+1)^p}

\frac{(n+1)n^p}{(n+1)^p}=\frac{n+1}{(1+\frac{1}{n})^p}

\lim \frac {U_n} {U_{n+1} } = \lim \frac{ n+1} { (1+ \frac {1} {n} )^p}= \infty

Which is >1 for values of p  

Hence by ratio test the series ∑Un convergent

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