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Proof of the preliminary test for infinite series

  1. Sep 2, 2013 #1
    1. The problem statement, all variables and given/known data
    Preliminary test: If the terms of an infinite series do not tend to zero, the series diverges. In other words if ##\lim_{ n \to \infty}a_n \neq 0## then the series diverges. But if the limit is 0 we have to test further.

    Suppose a series a series satisfy this condition, ##lim_{ n \to \infty}S_n= S##, and consequently ##\lim_{ n \to \infty}S_{n-1}=S##

    ##S_n-S_{n-1}=a_n##

    ##\lim_{ n \to \infty}(S_n-S_{n-1})=\lim_{ n \to \infty}S_n - \lim_{ n \to \infty}S_{n-1} = \lim_{ n \to \infty}a_n##

    ##S-S=\lim_{ n \to \infty}a_n=0##

    But if the terms of the series does not tend to 0, then this ##lim_{ n \to \infty}S_n=lim_{ n \to \infty}S_{n-1}=S## is not true, and the series cannot be convergent, or equivalently the series must diverge.
     
  2. jcsd
  3. Sep 2, 2013 #2

    vela

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    Did you have a question?
     
  4. Sep 2, 2013 #3
    The only task is to prove the test, and I just want to know if it's valid according to the PF members.
     
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