An infinite series is a ``sum'' of the form
Given a sequence
of numbers, the
partial
sum of this sequence is
We define the infinite series
by
For the sequence
, we have
.
Thus,
is divergent.
For the sequence
, we have
,
,
, etc.
In general,
for
odd and
for
even. Thus,
is divergent.
If one computes the partial sums for
one finds
,
,
,
,
,
,
. In fact,
, so
that
diverges, though we will
see why only later.
If one computes the partial sums for
, then one
obtains
,
,
,
,
,
.
In fact,
The series
is an example of a geometric
series. Computing, we find
,
,
,
,
. In fact,
.
A geometric series is a series of the form
In the above case
.
If
then
Subtracting from both sides, we obtain
Hence,
So
provided that the limit on the right exists.
If , then
, so that
If , then
does not exist, so
diverges.
Finally, in the case that , we have already seen that the series
diverges.
Compute the following sums
In the first case, we may write the sum as
.
So, the sum is
.
In the second case, we may write the sum as
so that this sum is
.