One of the most important things you will need to learn in this section
of the course is a list of standard examples of convergent and
divergent series. (The reasons for this will be clear when we get
on to discussing the comparison test for convergence.) And by far the
most important examples are the series
^{
}
^{-}

Geometric series.

The series
^{
}

**Proof.**

If 0.
By multiplying through by _{
=
=0
}_{
=
=0
=
=1
+1
}_{
=
0-
+1
}. If
0_{
=
0-
+1
1-
}^{
+1
}
, as required._{
0-0
1-=11-
}

If ^{
1
}
^{
}

If you prefer your series to start at ^{
}

In the next group of examples, we are not so lucky as to be able to give a nice formula for the limit. We can still analyse convergence using monotonicity, however. Recall that the exponent

Example.

The series
^{-1}

**Proof.**

We have the following inequalities

and so on:

Putting these together we have, for

which shows that the sequence of partial sums
is unbounded and hence
not convergent._{
=
=1
1
}

The series ^{-1}

Example.

For

**Proof.**

Suppose

and so on. This gives

Put =

-1

is positive. Then

so the sequence of partial sums, (, is bounded.
But as each term in the series is positive, (_{
)
} is
monotonic and hence convergent._{
)
}

The series ^{-}

Note that, in obvious contrast with ^{
}
^{-}