Proof of the preliminary test for infinite series

[Seydlitz]

Homework Statement

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.

 

[vela]

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[Seydlitz]

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The only task is to prove the test, and I just want to know if it's valid according to the PF members.