Comparison Test
This original textbook reading covers part of the AP topic. Follow the AP course outline for complete topic instruction and practice.
Section 3
Comparison Test
For the given series 
, let there be another series 
such that
<
- If converges, so does
- If diverges, so does
Conditions
For all n,
- an, bn > 0
- an < bn
Example 4.3.1
Use the comparison test to determine the convergence of the series.
Graph 4.3.1
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Converges or
Diverges?
Since the example series is less than the other series given, which is convergent, the example converges also.
Series Given for
Comparison
The series converges to 5
Example Series
The series converges to .3
Find a known series for comparison. In this example...
2n+5 > 2n
As the series is bounded above and is always increasing, it is therefore convergent.
Example 4.3.2
Use the comparison test to determine the convergence of the series.
Find a known series for comparison. In this example...
n-1 < n
The series diverges.
Graph 4.3.2
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Converges or
Diverges?
As the partial sums increase without bound, the series diverges.
Now, assuming that the series 
is convergent, this means that the partial sum sn is less than a finite value for every n, and that the partial sum sn has an upper bound.
Since the sequence of partial sums for sn, or
{sn}
is monotonically increasing and bounded above, this means that the sequence must converge. And according to Chapter 2, if a sequence of partial sums converges, so does its infinite series.
Therefore, the series 
converges.
Proof of Comparison Test
For the series and , the comparison test relies on the conditions that
- ak, bk > 0
- ak < bk
The proof makes use of the partial sums of the sequences, where
sn = 
and tn = 
Since the terms for both series are always positive,
sn < sn+1 and tn < tn+1
As such, the sequence of partial sums for both series are monotonically increasing.
And because ak < bk , this means that for every n, sn < tn .
Looking at the relationship between the partial sum tn and its infinite series,
tn = <
Under these conditions, it can now be said that
sn <
Chapter 4 Problems
Use the ratio, root, and comparison test to determine the convergence of the following series.
- The series diverges through the comparison test
Because ln n < n for n>2,
The series diverges.
- The series converges through the root test.
= 
= 
=
The series converges.
- The series converges through the root test.
=
=
=
=
=
The series converges.
- The series converges through the root test.
=
=
=
< 1
The series converges.
- The series converges through the comparison test.
Because 
The series converges.
- The series through the ratio test.
=
=
=
=
=
=
=
=
< 1
The series converges.
7. The series converges through the comparison test.
Because 
and 
...
The series is convergent.
- The series diverges through the ratio test.
=
=
=
=
=
The series diverges.
- The series converges through the comparison test.
Because 
The series converges.
- The series diverges through the ratio test.
=
=
=
=
The series diverges.
- The series converges through the ratio test.
=
=
=
=
=
The series converges.
- The series converges through the ratio test.
=
=
=
=
= 3
= 0 < 1
The series converges.
- The series diverges through the comparison test.
Because 
and 
The series diverges.
- The series converges through the ratio test.
=
=
=
=
=
< 1
The series converges.
- The series diverges through the root test.
=
=
=
=
The series diverges.
Glossary and deeper reading · 2 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Comparison tests
Comparison tests determine convergence by relating nonnegative terms to a known benchmark series. Direct comparison uses a correctly directed inequality, while limit comparison uses a positive finite limit of the term ratio. For example, 0≤1/(n²+1)≤1/n² shows convergence because the upper p-series converges. Being smaller than a divergent series or larger than a convergent one does not settle convergence.
-
Infinite series
An infinite series is the sum of the terms of an infinite sequence, defined through its partial sums. The series converges when those partial sums approach a finite value. For example, 1/2+1/4+1/8+⋯ converges to 1. Terms approaching 0 are necessary for convergence, but that condition alone is insufficient.