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Factorials within alternating series

  1. Jan 28, 2015 #1
    1. The problem statement, all variables and given/known data

    ∑ [ (-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.
     
  2. jcsd
  3. Jan 28, 2015 #2

    LCKurtz

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    Have you thought about whether the nth term of the series goes to zero?
     
  4. Jan 29, 2015 #3

    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 ?
     
  5. Jan 29, 2015 #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##?
     
    Last edited: Jan 29, 2015
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