Calculus 2
Test #3 Review Sheet
Page 1 of 7
Section 8.1: Sequences of Real Numbers
To graph a sequence on the TI-83:
(i).
Set the graphing mode to Seq using the MODE button.
(ii).
Enter the “general term” using the Y= button.
(iii).
Set the window properties using the WINDOW button.
(iv).
Graph the sequence using the GRAPH button.
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 which
0
an  L   for every n > N. If there is no such number L, then we say the sequence diverges.
(This definition is the starting point for proving Theorems 1.1, 1.2, and 1.3…)
Theorem 1.1: Limits of Combinations 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 that
0
0
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 
Definition 1.2: The factorial.
For any integer n > 1, the factorial n! is defined as the product of the first n positive integers.
n!  1 2  3   n
We define the factorial of zero to be one: 0!  1.
Calculus 2
Test #3 Review Sheet
Page 2 of 7
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.
Section 8.2: Infinite Series
Theorem 2.1: Sum of an Infinite Geometric Series.

a
For a  0, the geometric series  ar k converges to
if r  1 and diverges if r  1 . The number r
1 r
k 0
is sometimes called the common ratio or just the ratio.
Theorem 2.2: If you add up an infinite number of numbers and the sum doesn’t blow up, then
they must be really small numbers.

If
a
k 1
k
converges, then lim ak  0 .
k 
kth-Term Test for Divergence. (Contrapositive of Theorem 2.2)
If lim ak  0 , then
k 

a
k 1
k
diverges.
Theorem 2.3: Combinations of Infinite Series (are what you think they’d be).


If
a
k 1

k
converges to A, and
b
k 1
k

converges to B, then the series
a
k 1
k
 bk   A  B , and

  ca   cA for any constant c.
k 1

k

If
a
k 1

k
diverges, and
b
k 1
k

diverges also, then the series
a
k 1
k
 bk  diverges as well.
Calculus 2
Test #3 Review Sheet
Page 3 of 7
Section 8.3: The Integral Test and Comparison Tests

1
diverges.

k 1 k
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



a
f  x  dx and
i 1
1
k
either both converge or both diverge.
Corollary: p-Series

1
The p-Series  p converges if p > 1 and diverges if p  1.
i 1 k
Theorem 3.2: Error Estimate for the Integral Test
Suppose that f(k) = ak for all k = 1, 2, 3, …, where f is both continuous and decreasing, and

f(x)  0 for x  1, and that
 f  x  dx converges.
Then, the remainder Rn satisfies
1
0  Rn 


 a   f  x  dx .
k  n 1
k
n
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


If
a
k 1

a
k 1
k
converges, too.

k
diverges, then
b
k 1
k
diverges, too.
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, either
k  b
k

b
k 1
k
both converge or both diverge.

a
k 1
k
and
Calculus 2
Test #3 Review Sheet
Page 4 of 7
Section 8.4: Alternating Series

An alternating series is any series of the form
  1
k 1
k 1
ak  a1  a2  a3  a4  a5  a6 
Theorem 4.1: Alternating Series Test
If lim ak  0 and 0  ak 1  ak for all k  1, then the alternating series
k 

  1
k 1
k 1
ak converges.
Theorem 4.2: Error Bounds for the Partial Sum of an Alternating Series
If lim ak  0 and 0  ak 1  ak for all k  1, then the alternating series
k 

  1
k 1
k 1
ak converges to some
number S, and the error in approximating S by the nth partial sum Sn satisfies:
error  S  Sn  an1 .
Section 8.5: Absolute Convergence and the Ratio Test

a
An absolutely convergent series has the property that not only does
k 1
k
converge (the idea being

that the series contains negative terms which help it converge) , but
a
k 1

A conditionally convergent series has the property that
Theorem 5.1: Absolute Convergence Implies Overall Convergence


k 1

ak converges, then
a
k 1
k
converges.
Theorem 5.2: The Ratio Test

Given
a
k 1
(i).
(ii).
(iii).
k
, with ak  0 for all k, suppose that lim
if L < 1, the series converges absolutely.
if L > 1 (or L = ), the series diverges.
if L =1, no conclusion can be made.
k 
converges also.
 ak converges, but
k 1
If
k
ak 1
 L . Then:
ak

a
k 1
k
diverges.
Calculus 2
Test #3 Review Sheet
Page 5 of 7
Theorem 5.3: The Root Test

Given
a
k 1
(i).
(ii).
(iii).
k
, with ak  0 for all k, suppose that lim k ak  L . Then:
k 
if L < 1, the series converges absolutely.
if L > 1 (or L = ), the series diverges.
if L =1, no conclusion can be made.
Section 8.6: Power Series
Power Series Definition
A power series in powers of (x – c) is any series of the form

b  x  c
k 0
k
k
 b0  b1  x  c   b2  x  c   b3  x  c  
2
3
,
where the constants bk are called the coefficients of the series.
Theorem 6.1: Convergence of a Power Series

Given any power series  bk  x  c  , there are exactly three possibilities:
k
k 0
The series converges absolutely for all x  (-, ), and the radius of convergence is
r = .
(v).
The series converges only for x = c, and the radius of convergence is r = 0.
(vi).
The series converges absolutely for x  (c – r, c + r), for some radius of convergence r with 0 <
r < , and diverges for |x – r| > c. The endpoints of the interval need to be evaluated separately.
(iv).
Now we will look at what to do when someone else gives us a power series, and asks for what range of
x values it converges.
Primary Tool to Test for Convergence of a Power Series: The Ratio Test
Calculus 2
Test #3 Review Sheet
Page 6 of 7
Calculus Operations on a Power Series

If we define a function f(x) on the interval of convergence of a power series: f  x    bk  x  c  ,
k
k 0
then we can obtain the derivative and integral of the function by differentiating or integrating each
term of the power series.
d
2
3
f  x 
b0  b1  x  c   b2  x  c   b3  x  c  
dx


 b1  2b2  x  c   3b3  x  c  
2

  k  bk  x  c 
k 1
k 1
And
 f  x  dx    b
 b1  x  c   b2  x  c   b3  x  c  
2
0
3
 dx
1
1
1
2
3
4
 b0 x  b1  x  c   b2  x  c   b3  x  c  
2
3
4

1
k 1

 bk  x  c   K I
k 1 k  1
 KI
Section 8.7: Taylor Series
Taylor Series expansion of f(x) about x = c:

f k   c 
k
f  x  
 x  c
k!
k 0
MacLaurin Series: a Taylor series where c = 0.
Theorem 7.1: Taylor’s Theorem
Suppose that f has (n + 1) derivatives on the interval (c – r, c + r) for some r > 0. Then, for
x  (c – r, c + r), f(x)  Pn(x), the error (or remainder term) in using Pn(x) to approximate f(x) is:
n 1
f    z
n 1
Rn  x   f  x   Pn  x  
 x  c
 n  1!
For some number z between x and c.
Theorem 7.2:
Suppose that f has derivatives of all orders on the interval (c – r, c + r) for some r > 0, and that
lim Rn  x   0 for all x  (c – r, c + r). Then, the Taylor series for f expanded about x = c converges to
n
f(x); that is:
f k   c 
k
 x  c
k!
k 0
for all x  (c – r, c + r).

f  x  
Calculus 2
Test #3 Review Sheet
Page 7 of 7
Section 8.8: Applications of Taylor Series
1. Application #1: Finding approximate values of a function.
 Key: Pick an expansion point close to the value you want to compute,a nd just use the first
few terms.
2. Application #2: Conjecturing the values of a limit.
 Key: Plug in the Taylor series, then simplify.
3. Application #3: Approximating a definite integral.
 Key: Plug in the Taylor series, then integrate.
4. Application #4: Finding solutions to differential equations. This technique is called “the
Method of Froebenius” when applied to certain second-order differential equations.

 Key: Assume that a differential equation has a solution of the form y  x    bk x k .
k 0
Substitute this solution into the differential equation, collect like terms, then use the recursion
formulas you get to solve for the coefficients.
5. Application #5: Defining special functions, like the Bessel functions of order p, which are the
solutions of the differential equation x 2 y  xy   x 2  p 2  y  0 , where p is a nonnegative
 1
x2k  p
integer: J p  x    2 k  p
k ! k  p  !
k 0 2
6. Application #6: The Binomial Series.

k
Theorem 8.1: The Binomial Series

r
r
For any real number r, 1  x      x k for -1 < x < 1.
k 0  k 
This expands to:
r  r  1 2 r  r  1 r  2  3 r  r  1 r  2  r  3 4
r
r
x 
x 
x 
1  x   1  x 
1!
2!
3!
4!