This web page discusses one of the most powerful tests
for convergence of series of positive terms: the comparison test.
The main idea is, to determine if a series
_{
}_{
}
comparison test

itself
and the limit comparison test

which is is often slightly
easier to use but is just a slightly different form.

To apply ideas of monotonicity
from a previous web page
we shall assume all series here are series of *positive*
terms except where stated otherwise.

Comparison Test.

Suppose _{
}_{
}
__ _{
}__,

(a) If there is a positive _{
}_{
}
_{
}

(b) If there is a positive _{
}
_{
}
_{
}

**Proof.**

(a) Suppose that the conditions in (a) are satisfied and
_{
=
}
_{
}
_{
}
_{
}
for the
_{
=
=1
}__ _{
}__s.

By assumptions on

and therefore by monotonicity ( converges
as it is bounded. Hence _{
)
}_{
}

(b) Suppose that the conditions in (b) are satisfied. Then
by monotonicity the sequence of partial sums from _{
}
**
**
there is

By assumptions on

As was arbitrary, this means that
the sequence ( of partial sums is unbounded.
Hence _{
)
}_{
}

Examples are given in another page.

The calculations involved in finding constants

Limit Comparison Test.

Suppose _{
}_{
}
__ _{
}__,

as _{
}_{
}

**Proof.**

Suppose that _{
}_{
}
_{
}_{
}
_{
}_{
}
__ _{
<
}__ and

Now if _{
}
_{
}
_{
}__ _{
<
}__
for

Similarly, if _{
}
_{
}
_{
}_{
<
}
_{
}

Examples are given in another page.

This web page has introduced you to one of the most powerful techniques for proving (non)convergence of series consisting of positive terms: the comparison test. Strictly speaking you only need to learn the basic comparison test, but the limit comparison test is much more convenient to use in many situations.