BC 3
Comparison Test
Name: ____________________
At times it is difficult if not impossible to find a closed-form expression for the sequence of
partial sums. In this case we need to find other methods to determine whether a series converges
or diverges. For example, we already have the nth term test. Another useful test is the comparison
test.
Comparison Test: Suppose that 0  ak  bk for all k 

i.
If

k 1
bk converges, then

ii.
If
a
k 1
. Then

k


ak also converges and
k 1

k 1

ak   bk ; and
k 1

diverges, then
b
k 1
k
also diverges.
n
n
k 1
k 1
Proof: Note that the terms of each series are non-negative. Let Tn   bk and Sn   ak be the
associated sequences of partial sums. It follows immediately that Tn  Sn for all n .

i.
If
b
k 1
k
converges, then Tn converges, say to T 
. Then T is an upper bound for the
increasing sequence S n , and hence S n converges. Clearly lim Sn  T , so
n 
ii.
Use this theorem together with the fact that
1
n
k 1
k 1
diverges to show that the following two series
n 1
diverge. Explain each step clearly.


k 1

(2)


ak   bk .
Similar to convergence proof.

(1)

1
k
k 1
2
k
k
k 1
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Sequences and Series 3 p.1
Before the comparison test theorem can be used effectively, we would like to have a list of series

whose behavior is known. Since lim an  0   an diverges , we are only interested in series
n 
n 1
whose general term approaches 0. We can populate such a list by using the integral test which
we develop NOW.
What is an integral? It is the limit of a sum of rectangles as you get smaller and smaller
rectangles. What is the sum of an infinite series? It is the limit of partial sums as you take more
and more pieces. Should it be any surprise that these two different limits of sums should be
connected?
Draw the graph of any function f(x) which is defined and decreasing on [1, ), and which
has a horizontal asymptote along the x-axis.
1
2
3
4
5
6
7
Now, circumscribe rectangles about your curve with vertical edges at the integers. In other
words, draw left-hand Riemann rectangles starting at x = 1 and continuing indefinitely to the
right.
1) What is the height of the rectangle lying over the interval [1, 2]? What is its width? What is
its area? What about the rectangle lying over [2,3]? The rectangle over [n, n+1]?
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Sequences and Series 3 p.2
2) What is the total area of the rectangles between x = 1 and x = n + 1? Use sigma notation.
3) What is the total area under the original curve between x = 1 and n + 1? Use an integral.
4) How do the two expressions

n 1
1
n
f ( x) dx and
 f (k )
compare? Which is larger? Why?
k 1
5) What do you get when you take the limit of these expressions as n  ? State this result
carefully!
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Sequences and Series 3 p.3
Redraw the graph of f(x).
1
2
3
4
5
6
7
This time, inscribe rectangles under the curve with using right-hand Riemann rectangles
over the intervals [1, 2], [2, 3], etc.
6) What is the area of the rectangle lying over [n, n+1] this time?
7) Derive an expression for the total area under all the rectangles in your figure (infinitely many
rectangles).
8) What is the relationship between the area under the rectangles and the area under the curve
this time?
Note also that this time, the series is less than the integral. So we can make the series
either larger or smaller than the integral by keeping or leaving out the initial term of the series.
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Sequences and Series 3 p.4
This has two effects: 1) we can always use integrals to test series for convergence and
divergence, and 2) we can use the integral to make a guess as to the limit of the series if it
converges.
For the first observation, if the integral of the function diverges, we include the initial
term of the series, notice that the partial sums of the series are bigger than the “partial integrals”,
which diverge, so the series also diverges. If the integral of the function converges, we ignore
the initial term of the series temporarily, and the partial sums of the series are all less than the
integral. Since the function is positive, the partial sums are increasing and bounded by the value
of the integral, so that the whole series must converge. So we can have it either way.
To estimate the sum of a series from the integral, note the relationship


n 1


f (n)   f ( x) dx   f (n) . What this tells you is that the true sum of the series can’t be too
1
n2
much farther away from the integral of the function than the initial term of the series.
The Integral Test for Series: Let f(x) be a decreasing positive function on the interval [1, ).

Then the series
 f (n) converges if and only if the integral 

1
n 1
f ( x) dx converges. If both
converge,


1


f ( x) dx   f (n)  f (1)   f ( x) dx
1
n 1
What the integral test allows us to do is to take a series of decreasing positive terms and
use integrals that we already know how to do to help evaluate their convergence and divergence.
The sequence of general terms has to be positive and decreasing or the comparisons of whether
the function is larger or whether the rectangles are larger don’t work! (Draw some pictures and
see for yourself!).
Use the integral test to test the following series for convergence. For the ones that converge,
give a range inside which the sum of the series is guaranteed to lie.

1)
n
n1


1
2)
n
1
 n ln n
n 2

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Sequences and Series 3 p.5
We now have a three series to use in comparison to other series that we want to know about.
Let’s get a “few” more.
Use the integral test and/or comparison test to determine for which values of p does the series

1
converges/diverges.

p
n 1 n
Use all of the tools we have developed up to this point to determine whether each series
converges or diverges. State you reasons carefully and completely.


n
(1)  2
n1 n  1

(2)
n
n1
2
1
 sin n

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Sequences and Series 3 p.6

(3)

n1

1
n
(4)
ln n
n
2
n1



(5)

k0

(7)
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 4
3 
 5

k
1

  1  n 
n 1
(6)

k2
n

(8)

k 1
1
k 1
| sin k |
k2
Sequences and Series 3 p.7

(9)

k 2

1
k  2k

(8)
k0

(10) You should have determined that
1
 k 2
k 2
k
1
k!
converges. Find the value of this sum with an
error of less than .0001. Explain your error analysis.
(11) Find all values of x  0 for which the series


k 1
xk
converges.
k  2k
Here are some problems from the text that you will find enjoyable.
Pages 480-81: 5,9,10-12,23,25,43,44
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Sequences and Series 3 p.8