Calculus II Series - Things to Consider
Important: This cheat sheet is not intended to be a list of guaranteed rules to follow. This intro-
duces some hints and some ideas you may consider when choosing tests for convergence or divergence when
evaluating a given series.
It is usually a good idea to try using the Test for Divergence as a first step when evaluating a series
for convergence or divergence. If we can show that:
lim
n→∞
an6= 0
Then we can say that the series diverges without having to do any extra work.
Below are some general cases in which each test may help:
P-Series Test:
•The series be written in the form: P1
np
Geometric Series Test:
•When the series can be written in the form: Panrn−1or Panrn
Direct Comparison Test:
•When the given series, anlooks like a known, or more simple, series, bn
Limit Comparison Test:
•When you can see that the series looks like another convergent or divergent series, bn
•But it is hard to say whether bn> anor bn< an
Root Test:
•When the series can be written in the form: P(an)n
Alternating Series Test:
•When the series can be written in the form: P(−1)n+1anor P(−1)nan
Ratio Test:
•Whenever we are given something involving a factorial, e.g. n!
•Whenever we are given something involving a constant raised to the nth power, e.g. Pn+5
5n
Integral Test:
•If the sequence is:
–continuous
–positive
–decreasing (we can use the First Derivative Test here)
