\sum_(n=1)^(\infty ) (n!)/(5^(n))
You asked:
Evaluate the expression: \(\sum_{n = 1}^{\infty} {\frac{n!}{{5}^{n}}}\)
MathBot Answer:
The infinite series \(\displaystyle\sum_{n=1}^{\infty} \frac{n!}{5^{n}}\) diverges.
We can use the nth-term test to determine convergence of this series.
Given a series \(\displaystyle\sum a_n\), if \(\displaystyle\lim_{n \to \infty}a_n \neq 0\) or does not exist, then the series diverges.
This test cannot prove convergence.
\(\displaystyle\lim_{n \to \infty} 5^{- n} n!\) does not exist.
Therefore the series diverges.