Test for the convergence of the series $1/5 + 2!/5^2 + 3!/5^3+\ldots $

Question

The series is $1/5 + 2!/5^2 + 3!/5^3+\ldots $

It's $n$th term will be $n!/5^n $

The problem I'm facing that I'm getting infinity using D'Alembert's ratio test.

Answer 1

It is divergent. Notice that $n!$ grows faster than any exponential function.

Notice that computing a factorial requires you to multiply the previous answer by the next integer. But exponentiation requires you to always multiply the previous answer by the same integer. Hence at a certain point you will be multiplying the result of the factorial function by integers bigger than the base of the exponential function, always. So your sum will be summing up bigger and bigger terms, so it is divergent.