# Factorials within alternating series

ahmed markhoos

## Homework Statement

∑ [ (-1)^n * n!/(10^n) ]

2. The attempt at a solution

the problem is that I cannot use derivative to make sure that a(n) is decreasing neither L hopital rule to find the limit.

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## Homework Statement

∑ [ (-1)^n * n!/(10^n) ]

2. The attempt at a solution

the problem is that I cannot use derivative to make sure that a(n) is decreasing neither L hopital rule to find the limit.

Have you thought about whether the nth term of the series goes to zero?

ahmed markhoos
Have you thought about whether the nth term of the series goes to zero?

I don't know if this is correct or not! because I've used L' hopital rule for one side and left the side of n! without derivation

##\lim_{n\rightarrow \infty} {\frac{n!}{10^n}}##

## ln(f(n)) = ln{\frac {n!}{10^n}} ##
##ln(f(n)) = ln(n!) - ln(10^n)##
##ln(f(n)) = ln(n!) - n*ln(10)##

##\lim_{n\rightarrow \infty} {ln(n!) - n*ln(10)}##

Using L' hopital rule ##\lim_{n\rightarrow \infty} {ln(n!) - ln(10)}\ = ∞ ##

since: ##ln(f(n)) = ∞## $$f(n) = e^∞ = ∞$$

which make the series diverges, is this correct ?