# Proof of the preliminary test for infinite series

## 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.