Factorials within alternating series

  • #1
ahmed markhoos
49
2

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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  • #2
LCKurtz
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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?
 
  • #3
ahmed markhoos
49
2
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 ?
 
  • #4
LCKurtz
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That isn't the form for L'Hospital's rule and you certainly don't need L'Hospital's rule for this problem. Why don't you just note the terms are positive and check that they are increasing for large ##n##?
 
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