February Break Assignment - 2015
BC Calculus – Infinite Series and Convergence Tests
Part 1 – What are Sequences?
Sequences are lists of numbers that form a pattern, such as:
1, 3, 5, 7, 9, … , 2n + 1
1, 2, 4, 8, 16, 32, … , 2𝑛
or
They examples above are examples of two of the more common types of sequences we study in
math classes, arithmetic (change by the same amount) and geometric (change by a common ratio).
You won’t too many arithmetic sequences in BC, but geometric are a BIG DEAL.
It helps to think of sequences as the output values of a function with a domain of integers starting with
zero (n = 0, 1, 2, 3, …) or one. You could even make an input output table:
n
2𝑛
0
1
1
2
2
4
3
8
4
16
5
32
You won’t generally see sequences written this way, but it helps to place them in a context with which
you’re familiar.
Here are some more sequences. Take a look at them and try to determine a “general term” for each.
This is a term with “n” in it that could be used to find any specific term in the sequence.
1
(A)
1,
(B)
1,
(C)
1 , −2 ,
(D)
1,
2
1
2
,
,
1
4
1
3
1
1
4
,
,
,
1
3
1
9
,
1
8
1
4
1
,
16
1
,
5
, … , _________
, … , _________
1
1
,−4 ,
1
16
,
5
1
25
, … , _________
, … , _________
All problems are copyright College Entrance Examination Board and Education Testing Service
1997 BC #76
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February Break Assignment - 2015
Part 2 – Infinite Series
A series is found by adding together the terms of a sequence. To begin with, let’s look the series
based on the four examples from the previous page.
1 𝑛
(A)
∑∞
𝑛=0 (2) = 1 +
(B)
= 1+
(C)
= 1−
(D)
= 1+
1
2
1
2
1
2
1
4
+
+
+
+
1
4
1
1
3
−
1
9
4
1
4
+
1
+
8
1
+
3
1
+
+
+
1
16
16
1
5
+
[geometric series]
+ . . .
[harmonic series]
−. . .
[alternating harmonic series]
5
1
+ . . .
1
25
+ . . .
[p series]
Some of these series are special types that appear often in BC Calc. It can be helpful to rewrite
them in “sigma notation”. I’ve done (A) for you, please complete the rest yourself.
What we usually want to determine about a series is whether or not it converges to a specific
value. The rules and tests used to figure this out are lengthy and will take up a few weeks of the
course.
Using your calculator, try to determine whether each of the series above converges by
using computing terms and adding the results:
(A)
(B)
(C)
(D)
This process gives you an idea about whether certain series converge. However, this is not an
efficient way of testing for convergence, especially since you typically won’t have a calculator. We
will now explore the common types of series and the rules for convergence.
Part 3 – A Hierarchy of Convergence Tests
Although there is no strict ranking of the importance of convergence tests, the following is my
informal recommendation based on advice that I received and my own observations.
1.) The nth term Test for Convergence (a.k.a. “Kicking the Tires”*)
* - Credit Bill Scott for this term!
This rule states that if 𝑎𝑛 is the nth term of a series that:
If lim 𝑎𝑛 ≠ 0, then the series diverges.
𝑛→∞
2
If lim 𝑎𝑛 = 0, then the series may converge.
𝑛→∞
February Break Assignment - 2015
We start here because this test is typically quick, and allows us to know if it’s even worth continuing.
A way of stating this in everyday language is:
“If the individual terms of a series don’t get really really small, there’s no way that the series will
converge. And even then, it still may not.”
2.) Geometric Series
1 𝑛
The example of a geometric series from before is ∑∞
𝑛=0 (2) = 1 +
1
2
+
1
4
+
1
8
+
1
16
+ . . .
Recognizing and testing geometric series is important for two reasons:
1.) Their prevalence in the course and on the exam
𝑛
2.) They tend to be quickly recognizable - anything that can be rewritten in the form ∑∞
𝑛=0 𝑎𝑛 (𝑟)
where 𝑟 is a constant is a geometric series.
You should have found that the series above converges. The rule for convergence of a geometric
series is:
Define the “a” to be the first term and “r” to be the common ratio.
If |𝑟| < 1 then the series converges, and the sum is:
𝑎
𝑆𝑛 = 1−𝑟
Geometric Series AP Multiple Choice Problems:
1997 BC #14
2003 BC #10
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February Break Assignment - 2015
3.) P-series:
The rule for p-series is:
𝑘
If a series can be written in the form ∑∞
𝑛=1 𝑛𝑝 , then:
If 𝑝 ≤ 1, then the series diverges
If 𝑝 > 1, then the series converges
The Harmonic Series – a special case of the “p-series”
1
The example of the harmonic series from above is ∑∞
𝑛=1 𝑛 = 1 +
series comes up frequently, and it diverges.
1
2
+
1
3
+
1
Another p-series that came up earlier in this handout was ∑∞
𝑛=1 𝑛2 = 1 +
Does this series converge?
1
4
1
4
+
+
1
5
1
9
+ . . . . This
+
1
16
+
1
25
+ . . .
2003 BC #22
4.) Alternating Series Test:
𝑛∙
If the series can be written in the form ∑∞
𝑛=1(−1) ∙ 𝑏𝑛 , then
If |𝑏𝑛+1 | < |𝑏𝑛 | for all n above a finite number, then the series converges
The process of alternating is very powerful in terms of convergence, which is one of the reasons
why we sometimes discuss the idea of absolute and conditional convergence.
∞
A series ∑∞
𝑛=1 𝑎𝑛 is called conditionally convergent if it converges, but ∑𝑛=1|𝑎𝑛 | does not.
It is called absolutely convergent if ∑∞
𝑛=1|𝑎𝑛 | does converge.
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February Break Assignment - 2015
The Alternating Harmonic Series
The example of the alternating harmonic series from above is
∑∞
𝑛=1
(−1)𝑛−1∙
𝑛
=1−
1
2
+
1
−
3
1
+
4
1
5
−. . .
.
This series comes up frequently, and it converges.
Here are some multiple choice questions that utilize multiple convergence tests:
1998 BC #18
1998 BC #76
2003 BC #24
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February Break Assignment - 2015
5.) Other Convergence tests:
a.)
Telescoping Series:
1
1
If the series can be written as partial fractions, such as ∑∞
𝑛=1 𝑛 − 𝑛+1, then is called at
telescoping series. To test for convergence, write out terms and see if they cancel to leave a
finite number.
This name is a reference to the old collapsible telescopes.
b.)
Comparison Tests:
There are two comparison tests, which students frequently confuse. What they share in
common is that both tests start with a series that is unknown and compare to a series whose
convergence is well known. They just go about it in different ways.
i.)
Direct Comparison
∞
If you know that a series ∑∞
𝑛=1 𝑎𝑛 converges, and each term of the series ∑𝑛=1 𝑏𝑛 is less
∞
than the corresponding term in ∑∞
𝑛=1 𝑎𝑛 , then ∑𝑛=1 𝑏𝑛 also converges.
∞
If you know that a series ∑∞
𝑛=1 𝑎𝑛 diverges, and each term of the series ∑𝑛=1 𝑏𝑛 is greater
∞
than the corresponding term in ∑∞
𝑛=1 𝑎𝑛 , then ∑𝑛=1 𝑏𝑛 also diverges.
One way of explaining this to students is to provide a graph, such as…
x



6

February Break Assignment - 2015
1
1
∞
The graph is a representation of the series ∑∞
𝑛=1 𝑛2 and ∑𝑛=1 𝑛2 +1. Since we already know that
1
1
∞
∑∞
𝑛=1 𝑛2 converges, were using it as a comparison for ∑𝑛=1 𝑛2 +1. Notice how the function on the top
“squeezes” the lower function towards a finite number.
1
1
1
1
∞
So if ∑∞
𝑛=1 𝑎𝑛 = ∑𝑛=1 𝑛2 = 1 + 4 + 9 + 16 + ⋯ and
1
∞
∑∞
𝑛=1 𝑏𝑛 = ∑𝑛=1 𝑛2 +1 =
1
2
1
1
∞
and ∑∞
𝑛=1 𝑎𝑛 converges (by the p-series test), then ∑𝑛=1 𝑏𝑛 must converge as well, since
1
<1
2
1 1
<
5 4
1
1
<
10 9
1
1
<
17 16
2008 BC #82
ii.)
Limit Comparison
∞
If you know that a series ∑∞
𝑛=1 𝑎𝑛 converges, and the ratio of the general terms for ∑𝑛=1 𝑎𝑛
and ∑∞
𝑛=1 𝑏𝑛 approach a positive constant, then the series both behave the same way
(convergent or divergent).
Or, if lim
𝑎𝑛
𝑛→∞ 𝑏𝑛
∞
=k, 0 < 𝑘 < ∞, then ∑∞
𝑛=1 𝑎𝑛 and ∑𝑛=1 𝑏𝑛 either both converge or diverge.
7
1
+ 5 + 10 + 17 + ⋯ ,
February Break Assignment - 2015
c.)
Integral Test
You may want to teach the integral test in conjunction with improper integrals, since the test relies
upon them.
∞
If 𝑓(𝑛) = 𝑎𝑛 , then ∫1 𝑓(𝑥)𝑑𝑥 and ∑∞
𝑛=1 𝑎𝑛 either both converge or both diverge.
In other words, if you could rewrite the general term of the series as a function in terms of x, then
the improper integral of the function will behave the same way as the series (though they will not
necessarily converge to the same value.
Here’s an example of a series where you would probably need to use the integral test to test for
convergence:
∞
∑
𝑛=1
ln(𝑛)
𝑛2
In general, series involving ln(𝑛) are good candidates for the integral test.
1998 BC #22
8
February Break Assignment - 2015
2008 BC #79
d.)
Ratio Test
The ratio test is most frequently associated with finding the radius of convergence of a power
series (another topic). However, it can be used on some infinite series.
Given the series ∑∞
𝑛=1 𝑎𝑛 :
If lim |
𝑛→∞
𝑎𝑛+1
|<1,
𝑎𝑛
the series converges
𝑎𝑛+1
|>1,
𝑛→∞ 𝑎𝑛
If lim |
If lim |
𝑛→∞
𝑎𝑛+1
|=1,
𝑎𝑛
the series diverges
the test is inconclusive
Series with a factorial (n!) are good candidates for the ratio test.
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February Break Assignment - 2015
2008 BC #4
Summary

Sequences are lists of numbers that form patterns.

Series are the summation of terms of a sequence. Infinite series are series with infinitely
many terms.

Some series converge to finite numbers, even if they contain infinitely many terms (wow!).
We test if a series converges using several convergence tests. One can attack series
problems strategically by thinking of the tests as follows:
1.
2.
3.
4.
5.
nth term test – is it even worth continuing?
geometric series – come up frequently
p-series – including the harmonic series, which is important
alternating series test – a relatively easy test that also comes up a lot
The rest – telescoping, comparison tests, integral test and ratio test*
* = very important in another context – intervals of convergence
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