Consider the following sequence of partial sums:
Can we compute
These partial sums look as follows:
It looks very likely that
, if it makes
any sense. But does it?
In a moment we will define
A little later we will show that
, hence indeed
(Sum of series)
is a sequence, then the sum of the series
provided the limit exists. Otherwise we say that
Consider the geometric series
To see this, multiply both sides by
that all the terms in the middle cancel out.
For what values of
, it's clear since
series also diverges (since the partial sums are
For example, if and
, we get
as claimed earlier.