nth Term Divergence Test
Consider a series
∞
X
n=1
an. If lim
n→∞
an6= 0 then the series diverges.
USED: When you suspect the terms of the given series do not approach zero. This is a quick and
straightforward test, assuming the limit of the terms is a manageable computation. Understand
how the behavior of the terms can determine divergence of the series. Generally, this test is helpful
when the series seems a bit “oddball” in form or is not a more natural candidate for another
convergence test.
NOTE: This test is inconclusive if the terms do approach zero. If lim
n→∞
an= 0, then you have
more work to do with a different convergence test. Simply put, if the terms do approach 0
then you know nothing except that there is posssible convergence. Remember, the Harmonic Series
∞
X
n=1
1
nis a divergent series even though the terms 1
nshrink to zero as n→ ∞.
WARNING: Do not create an nth term convergence test. There is no such result or test. Why?
WARNING: Do not declare convergence from a divergence test.
APPROACH:
•Given a series, step aside and examine the terms of the series.
•If you see that the terms will approach zero as napproaches infinity, then do not waste your
time proving that the terms approach zero. This test will not help you. Move onto another
appropriate convergence test.
•If you suspect that the terms do not go to zero as napproaches infinity, then carefully
compute lim
n→∞
an. Show the actual limit answer, AND write that it does not equal zero.
That is the condition of the test you need to write clearly. Use all of your limit training for
sequences. If you need L’H Rule, then make sure that you switch to the related function
having xas the variable.
•Think about this diveregnce test in the following way... If lim
n→∞
an=L6= 0, then it says
that the sequence terms {an}∞
n=1 converge to the limit L. However, because the terms do not
converge to 0, then the original series diverges.
