Now that we’ve got all of our tests out of the way it’s time
to think about organizing all of them into a general set of guidelines to help
us determine the convergence of a series.
Note that these are a general set of guidelines and because
some series can have more than one test applied to them we will get a different
result depending on the path that we take through this set of guidelines. In fact, because more than one test may
apply, you should always go completely through the guidelines and identify all
possible tests that can be used on a given series. Once this has been done you can identify the
test that you feel will be the easiest for you to use.
With that said here is the set of guidelines for determining
the convergence of a series.
Again, remember that these are only a set of guidelines and
not a set of hard and fast rules to use when trying to determine the best test
to use on a series. If more than one
test can be used try to use the test that will be the easiest for you to use
and remember that what is easy for someone else may not be easy for you!
Also just so we can put all the tests into one place here is
a quick listing of all the tests that we’ve got.
Divergence Test
If then will diverge
Integral Test
Suppose that is a positive, decreasing function on
the interval and that then,
- If is convergent so is .
- If is divergent so is .
Comparison Test
Suppose that we have two series and with for all n
and for all n. Then,
- If is convergent then so is .
- If is divergent then so is .
Limit Comparison Test
Suppose that we have two series and with for all n. Define,
If c is positive (i.e. ) and is finite (i.e. ) then either both series converge or
both series diverge.
Alternating Series
Test
Suppose that we have a series and either or where for all n. Then if,
- and,
- is eventually a decreasing sequence
the series is convergent
Ratio Test
Suppose we have the series . Define,
Then,
- if the series is absolutely convergent (and
hence convergent).
- if the series is divergent.
- if the series may be divergent,
conditionally convergent, or absolutely convergent.
Root Test
Suppose that we have the series . Define,
Then,
- if the series is absolutely convergent (and
hence convergent).
- if the series is divergent.
- if the series may be divergent,
conditionally convergent, or absolutely convergent.