Calc 2 Lecture Notes
Section 8.3
Page 1 of 5
Which test do I use for a sequence or series?
Are you evaluating a sequence or a series (infinite sum)?
If a series, go to page 3.
Otherwise…
For a Sequence
Calculate a few terms
Graph some terms
Definition 1.1: Convergence of a sequence and divergence of a sequence.

The set an n  n converges to L if and only if given any number  > 0 there is an integer N for
0
which an  L   for every n > N. If there is no such number L, then we say the sequence
diverges.
Theorem 1.1: Combinations of limits of sequences.


If an n  n and bn n  n both converge, then
0
0
(i).
lim  an  bn   lim an  lim bn
(ii).
lim  an  bn   lim an  lim bn
(iii).
lim  anbn   lim an
(iv).
 a  lim an
lim  n   n
(assuming lim bn  0 ).
n 
n  b
bn
 n  lim
n 
n
n
n
n
n 
n 

n 
n 
 lim b 
n
n 
Theorem 1.2: The limit of a sequence is the limit of its function.
If x  and lim f  x   L , then lim f  n   L for n .
x 
n
Note that the converse is not true. Counterexample: lim cos  2 n 
n 
Theorem 1.3: Squeeze Theorem


Suppose that an n  n and bn n  n both converge to the limit L. If there is an integer n1  n0 such
0
0
that all n  n1 guarantees that an  cn  bn, then cn n  n converges to L as well.

0
Corollary 1.1:
If lim an  0 , then lim an  0 as well.
n 
n 
Calc 2 Lecture Notes
Section 8.3
Page 2 of 5
Definition 1.3: Increasing and Decreasing sequences.

The sequence an n 1 is increasing if a1  a2  a3   an  an1 
The sequence an n 1 is decreasing if a1  a2  a3   an  an1 
If a sequence is either increasing or decreasing it is called monotonic.

Key trick: To determine whether a series is monotonic, look at the ratio of successive terms.
Definition 1.4: Bounded sequences.

The sequence an n 1 is bounded if there is a number M > 0 (called a bound) for which |an| < M
for all n.
Theorem 1.4: Convergence of bounded monotonic sequences.
Every bounded, monotonic sequence converges.
Calc 2 Lecture Notes
Section 8.3
Page 3 of 5
For a Series
Get a “feel” for the series:
1. Evaluate a few partial sums.
2. Graph the first few partial sums.
3. Evaluate some large-n partial sums.
If the terms are always positive:
1. Quick check for convergence: can you work out the answer to the sum?

1
10
k

a. Geometric series: example:   0.1 
(Section 8.2)
1  0.1 9
k 0

a
ar k converges to
if r  1 and diverges if r  1

1 r
k 0
b. Telescoping series:


1
1
1
1 1   1 1   1 1 
 
              1 (Section 8.2)

k 1  1 2   2 3   3 4 
k 1 k  k  1
k 1 k
2. Quick check for convergence: is the series a p-Series?

1
a. The p-Series  p converges if p > 1 and diverges if p  1. (Section 8.3)
i 1 k
3. Quick check for divergence: do the terms grow instead of tending to zero?
a.  divergence by the kth term test. Example:

 k   because lim a
k 
k 1
k
0.
(Section 8.2)

b. The harmonic series diverges:
1
 k   . (Section 8.2)
k 1
4. Develop an intuition for whether it will converge or diverge, then compare the sequence
to a known converging or diverging integral or series:
a. Theorem 3.1: The Integral Test for Convergence of a Series If f(k) = ak for all
k = 1, 2, 3, …, and f is both continuous and decreasing, and f(x)  0 for x  1, then


f  x  dx and
1

a
i 1
k
either both converge or both diverge.
b. Theorem 3.3: The Comparison Test for Convergence of a Series Suppose that
0  ak  bk for all k . If

 bk converges, then
k 1

 ak diverges, then
k 1

a
k 1
k
converges, too. If

b
k 1
k
diverges, too.
c. Theorem 3.4: The Limit Comparison Test for Convergence of a Series
a
Suppose that ak, bk > 0, and that for some finite number L, lim k  L  0 . Then,
k  b
k

either
a
k 1

k
and
b
k 1
k
both converge or both diverge.
Calc 2 Lecture Notes
Section 8.3
Page 4 of 5
If the terms of the series are not always positive, including if they alternate sign:
1. If the terms do go to zero, and they alternate sign, then the series converges. (Section
8.4)

2. If some of the terms are negative, but
a
k 1
k
converges (which you would test using
techniques from the previous page), then the series converges absolutely (Section 8.5)

3. Theorem 5.2: The Ratio Test Given
a
k 1
k
, with ak  0 for all k, suppose that
ak 1
 L . Then:
k  a
k
a. if L < 1, the series converges absolutely.
b. if L > 1 (or L = ), the series diverges.
c. if L =1, no conclusion can be made.
lim

4. Theorem 5.3: The Root Test Given
a
k 1
k
, with ak  0 for all k, suppose that
lim k ak  L . Then:
k 
a. if L < 1, the series converges absolutely.
b. if L > 1 (or L = ), the series diverges.
c. if L =1, no conclusion can be made.
Calc 2 Lecture Notes
Section 8.3
Test
Geometric
Series
When to Use
Kth-Term Test
All series.
Integral Test

 ar
Comparison
Test
Conclusions
a
Converges to
if r  1 and
1 r
diverges if r  1 .
k
k 0
If lim ak  0 , the series diverges.
k 



k 1
1
 ak , where ak  f  k  , f
 ak and
is continuous and
decreasing, and
f(x)  0

1

p
k 1 k
0  ak  bk for all k .
converge or both diverge.
k 1
p-Series
Page 5 of 5

b
k 1
8.3
8.3
converges, then
k
8.2
8.3
 f  x  dx both
Converges if p > 1 and diverges if
p  1.
If
Section
8.2

a
k 1
converges.
k

If
a
k 1
k
diverges, then

b
k 1
Limit
Comparison
Test
ak, bk > 0, and
a
lim k  L  0
k  b
k
Alternating
Series Test
  1
Absolute
Convergence
Ratio Test
Root Test

k 1
k 1
ak , where
ak  0
Series with some positive
and some negative terms
(including alternating
series)
Any series (especially
those with exponentials or
factorials)
Any series (especially
those with exponentials)
k
diverges.

 ak and
k 1

b
k 1
k
both converge or
8.3
both diverge.
If lim ak  0 and ak 1  ak , then the 8.4
k 
series converges.

If
a
k 1

k
converges, then
a
k 1
8.5
k
converges absolutely.
ak 1
 L and if L < 1, the
k  a
k
series converges absolutely.
8.5
If lim k ak  L and if L < 1, the
8.5
If lim
k 
series converges absolutely.