Convergence Tests
1) Divergence Test
If lim ? 5 Á !, then !? 5 diverges.
5Ä_
NOTE: If
lim ? 5 œ !, then !? 5 may or may not converge.
5Ä_
2) Integral Test
_
Let ! ? 5 be a series with non negative terms, and let 0 aBb be the function that results when 5 is replace
_
by B in the formula for ? 5 . If 0 aBb is non increasing and continuous for B " , then ! ? 5 and '" 0 aBb
5œ"
_
5œ"
both converge or diverge.
NOTE: Use when 0 aBb is easy to integrateÞ
3) Comparison Test
Let !+ 5 and !, 5 be series with non negative terms such that + 5 Ÿ , 5 .
If !, 5 converges, !+ 5 , and if !+ 5 diverges, !, 5 divergesÞ
NOTE: Use this test as a last resort. Some other tests are easier to use.
4) Limit Comparison Test
Let !+ 5 and !, 5 be series with non negative terms such that 3 œ lim
+5
5Ä_ , 5
If !  3   _, then both series converge or diverge. If 3 = 0 or 3 = _, the test fails.
NOTE: Require some skill in choosing the series !, 5 for comparison.
5) Alternating Series Test
5
5"
If + 5  ! for all 5 , the series !a  "b + 5 or !a  "b + 5 converges if
a) + 5 is non increasing
5
b) lim + 5 œ !
5
5Ä_
Definition of Absolute Convergence
Let !+ 5 have some positive and some negative terms. If ! l+ 5 l converges then !+ 5 converges
absolutely (in absolute value). If ! l+ 5 l diverges but !+ 5 converges then !+ 5 converges conditionally.
6) Ratio Test
Let !? 5 be any series such that 3 œ lim
|? 5 " |
5Ä_ |? 5 |
a) the series converges absolutely if 3  ".
b) the series diverges if 3  " or 3 œ _.
c) No conclusion if 3 œ ".
Note: Try this when ? 5 involve factorials and 5 >2 powers
7) Root Test
5
Let !? 5 be any series such that 3 œ lim È
|? 5 |
5Ä_
a) the series converges absolutely if 3  ".
b) the series diverges if 3  " or 3 œ _.
c) No conclusion if 3 œ ".
Note: Try this when ? 5 involve 5 >2 roots.