is called the infinite series with as its general term.

if the sequence of partial sums converges, its limit is called the sum of the series.

*Denote
the first term of the geometric series and by
its ratio; according to Prop. 4.2,
we have:
. thus:
*

- If , the series is convergent and its sum is equal to .
- If , the series is divergent.

Hence:

It follows that the series . is convergent and its sum is equal to 1.

Take the series:

Despite the fact that the general term of the series has a limit equal to 0, this series is divergent (it has an infinite sum).

We prove that this series is divergent by grouping the terms in the following way:

Noah Dana-Picard 2007-12-28