Math 461
Sequences and Series in R
(Review from Calculus III -- see Stewart Chapter 12)
The following homework exercises are due Thursday 12-2-10.
A. Sequences.
Definition. A sequence in R is a function from the natural numbers {1, 2, 3, ...} to R. We
usually write the function values with subscripts instead of function notation. For example,
instead of writing f (n) = 1/n, we would write: an = 1/n. The entire sequence is denoted with
parentheses: (a n ) . A single term is denoted without parentheses: If an = 1/n then a5  15 .
Definition. Let L be a real number. A sequence (a n ) in R has the limit L provided that for every
  0 , there exists a corresponding integer N such that for all n > N, | an  L |   . If a sequence
has a real limit L, it is said to be convergent, and we write lim a n  L . If there is no real number
n 
L such that (a n ) converges to L, the sequence is divergent.
Exercise 1. Let (a n ) be a sequence in R. Prove: If lim | a n |  0 then lim a n  0
n 
n 
Definition. A sequence (a n ) in R is bounded if there exist real numbers a and b such that a  b
and for all n, a n  [a, b] .
Exercise 2. For each condition below, give an example of a specific sequence in R which
satisfies the condition.
a) The sequence converges to 0, and all the terms an are positive.
b) The sequence converges to 0, and all the terms an are negative.
c) The sequence converges to 0, and the terms alternate between positive and negative.
d) The sequence converges to 5, and all the terms are less than 5.
e) The sequence is bounded and divergent.
f) The sequence is unbounded and divergent.
Exercise 3. Let L be a real number. Let (a n ) be a sequence in R. Prove: If lim a n  L then
n 
(a n ) is a bounded sequence.
1|Page
Definition. A sequence (a n ) in R diverges to infinity provided that for every M > 0, there is an
integer N such that for all n > N, an > M. In this case, we write lim a n   .
n 
Definition. A sequence (a n ) in R diverges to negative infinity provided that for every M < 0,
there is an integer N such that for all n > N, an < M. In this case, we write lim a n   .
n
Exercise 4. For each condition below, give an example of a specific sequence in R which
satisfies the condition.
a) Sequence diverges to infinity.
b) Sequence diverges to negative infinity.
c) Sequence diverges, but lim a n   and lim a n   .
n 
n
Limit Laws for Sequences. Let (a n ) and (bn ) be convergent sequences in R, and c  R .
a) lim c  c
n 
b) lim (a n  bn )  lim a n  lim bn
n 
n 
n 
c) lim (a n  bn )  lim a n  lim bn
n 
n 
n 
d) lim (a n  bn )  lim a n  lim bn
n 
n 
n 
e) If lim bn  0 , then lim (a n / bn )  lim a n / lim bn .
n 
n 
n 
n 
f) If p > 0 and an  0 then lim (an )  (lim an ) p .
p
n
n
Squeeze Theorem for Sequences. Let n 0 be a positive integer. Let (a n ) , (bn ) , and (c n ) be
sequences in R with lim a n  L and lim c n  L , where L  R . If an  bn  cn for all n ≥ n 0 ,
n 
n 
then lim bn  L .
n 
sin( n / 2) 

Exercise 5. Use the squeeze theorem to find the limit of  3 
.
n


Definition. (a n ) is increasing if for all n, an  a n1 . (a n ) is non-decreasing if for all n,
an  a n1 .
Definition. (a n ) is decreasing if for all n, an  an1 . (a n ) is non-increasing if for all n,
an  a n1 .
2|Page
Definition. (a n ) is monotonic if (a n ) is either increasing or decreasing.
Monotonic Sequence Theorem. Every bounded, monotonic sequence in R is convergent.
Exercise 6. For each sequence below, determine (i) if the sequence converges (if so, find the
limit); (ii) if the sequence is monotonic (if so, state "increasing" or "decreasing"); (iii) if the
sequence is bounded (if so, find a closed interval containing all the terms of the sequence).
Justify each answer.
n 1
a) a n 
3n  1
n
b) bn 
1 n
c) cn  cos( 2 / n)
d) d n 
ln( n)
,n2
ln( 2n)
e) en  ln( n  1)  ln( n)
f)
fn 
sin( 2n)
1 n
g) g n  sin( n)
h) (hn )  ( 2 , 2 2 , 2 2 2 ,...)
B. Series
If we try to add the terms of an infinite sequence, we get an expression of the form
a1  a2  a3   which is called an infinite series (or just a series) and is denoted by the symbol

a
n 1
n
or, more briefly,
a
n
.
For some sequences, it may not make sense to try to add all the terms. For example, if
a n  n 2 , then  an = 1 + 4 + 9 + 16 + ... which just goes off toward infinity.
For other sequences, it does seem to make sense to add all the terms and produce a real
1
number result. For example, if a n  n , then  an = 1/2 + 1/4 + 1/8 + 1/16 + ... . Since
2
3|Page
1 1 1
1 2 n 1  1
, it seems that as n
   n 
2 4 8
2
2 n 1
approaches infinity the sum will approach 1. We make this idea precise by defining the partial
sums of a sequence.
1/2 + 1/4 = 3/4, and 1/2 + 1/4 + 1/8 = 7/8, and

Definition. Given a series
 a n , its nth partial sum is sn =
n 1
n
a
i 1
i
. If the sequence of partial

sums ( s n ) is convergent to some real number s, then the series

we write
 a n = s. That is, if the limit exists then
n 1

 an  lim
n 1
n 
a
n 1
n
is called convergent and
n
a
i 1
i
. The number s is called the
sum of the series. If there is no such limit for the partial sums, we say the series is divergent.
Exercise 7. Calculate the first 5 partial sums for each series, and try to decide whether the series
converges or diverges.

a)
 (1)
n 1
n 1

b)
 ( 0 .9 )
n
n 1

c)
1
n
n 1
(1) n 1

n
n 1

d)

e)
1
 n!
n 1
If | a n | does not approach zero as n increases, there is no hope for the corresponding series

a
n 1
n
to be convergent.
Divergence Test. If | a n | is divergent, or converges to a real number other than 0, then the series

a
n 1
n
diverges.
4|Page
Exercise 8. Use the divergence test to prove that the following series diverge.

a)
 (1)
n 1
n 1

b)
 (1.1)
n
n 1

c)
 cos( n)
n 1
Definition. If there exist real numbers a and r such that
say
a
n
a
n
= a  ar  ar 2    ar n   , we
is a geometric series with ratio r.
Geometric Series Test. The geometric series with ratio r is convergent iff | r | < 1. In this case,
a
the sum of the series is a  ar  ar 2    ar n   
.
1 r
Exercise 9. Find a geometric series with ratio 1/3 which converges to the sum s = 5.
Exercise 10. State each of the following tests (consult your calculus book), and give a specific
example for each.
a) Integral Test
b) p-series Test
c) Comparison Test (for showing convergence)
d) Comparison Test (for showing divergence)
e) Limit Comparison Test
f) Alternating Series Test
g) Absolute Convergence Test
h) Ratio Test
i) Root Test
Exercise 11. What does it mean for a series to be absolutely convergent? To be conditionally
convergent? Give a precise definition for each term, and give a specific example for each.
Exercise 12. Can a series be conditionally convergent if all its terms are positive? Explain.
5|Page