AP Calculus BC
Infinite Series Tests Review
You should understand the following key ideas for the quiz next Tuesday:






l’Hopital’s Rule for indeterminate limits of the form 00 or 

Improper integrals
Geometric series and the condition for convergence
Divergence test for infinite series
Direct comparison test, integral test, ratio test, alternating series test
Error in approximating an alternating series
This chart may help organize your thinking about the tests:
Test
Divergence
(or nth
Term)
Geometric
Series
Telescoping
Series
Series
u
n 1
lim un  0
n 
n

 ar
r 1
n

 u
n 1
n
 un 1 

lim un  L
  1
p 1
p
n 1
n 1
Remainder:
RN  uN 1
n 

u
n 1
n
, with
un  f  n   0

 f  x  dx converges
1

u
n 1
Direct
Comparison
Test
p 1
0  un1  un
and lim un  0
un

u
n 1
n

Remainder:
1
0  RN   f  x  dx
 f  x  dx diverges
u
lim n1  1
n  u
n
0  un  cn where
0  dn  un where

 cn converges
n 1

N
u
lim n 1  1
n  u
n
n
Comment
This test cannot be
used to show
convergence
a
S 
1 r
S  u1  L
n 
1
n
n 1
Ratio Test
r 1
n0

Integral
Test
(f is
continuous,
positive, and
decreasing)
Diverges

p-series
Alternating
Series
Converges

d
n 1
n
diverges
Test is inconclusive
u
if lim n1  1
n  u
n
Some Review Questions...
1.
Test for convergence or divergence using any appropriate test.

(a)

 1


 
n 1  4 


(g)


5
(b)
n
n 1
(d)
n 1
n
n
2
n 1
3n2
n
n 1
(k)
(o)

n 1
 1
(n)
n
(p)
n 2n
n
n
2
1
10n  3
n2n
n 1

10
(i)
3 n3


cos n

n
n 1 2
(l)
 1
n
 n ln n
n 1

ln n

2
n 1 n
(n)

 1
n

(q)
n
 3
n
 2 n!
n 1
n
2.
You are told that the terms of a positive series appear to approach 0 rapidly as n approaches
infinity. In fact, u7  0.0001 . Given no other information, does this imply that the series
converges? Support your answer with examples.
3.
Use the ratio test


n 1
4.
 1
n
n 1
n
3n 1
n!
n 1
 3

n 1 3  5  7   2 n  1

3n
3
 2n
n 1

n7 n

n 1 n !


(f)

2n

2
n 1 4n  1

(m)

n
n 1
2n

n 1 n  1

(h)

(j)
(c)

(e)
 1

5

n 1 n
for convergence.
Hopefully you saw that the ratio test failed for the series in the previous question. Since it is an
alternating series, if you can show that un1  un , then the sequence is decreasing. By letting
x
, use calculus to find the derivative and demonstrate that f   x   0 (i.e. it is
x 1
decreasing!) for all x  1 . Then, complete the rest of the Alternating Series Test to show the
series converges.
f  x 
5.
What is the minimum number of terms needed of the series from question 3 to approximate the
actual sum to an accuracy of less than one-millionth?