AP Calculus BC
Mr. McMullin
Unit: 8: Sequences, Series, and Convergence Tests
Lesson: 1 Class Date ____________________
Learn how to / Learn about: Sequences

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
Sequence – a function whose domain is a set of integers, usually all positive integers.
General term
Recursive sequence – terms are defined in term of previous term(s).
Limit of a sequence, convergent sequence, and divergent sequence.
Sequence defined by a function

Geometric sequence: an  a1r n 1 , n  1, 2,3,5,...



Limit laws of sequence (Box R p, 541)
Squeeze Theorem for Sequences
Bounded sequence; a bounded monotonic sequences converges to its least upper
bound. Convergent sequences are bounded.
If a sequence is equal to a function at the integer value, then they function and the
sequence have the same limit.

Sections / pages
1
F § 9.4
R § 10.1
Hand-in Homework
R: p. 546 – 3, 6, 7, 8, 13, 15, 17, 20, 28, 37, 43, 50, 63, 67
Show work
Due date
Lesson: 2 Class Date ____________________
Learn how to / Learn about: Summing infinite series

Sequence of partial sums of a series:
N 
 an    a1  ,  a1  a2  ,  a1  a2  a3  ,
 n1 

Sigma notation
,  a1  a2  a34 
 aN 

An (infinite) series is a single number defined as the limit of the sequence of partial sums of a

a
sequence.
n 1
n
N 
 lim  an 
N 
 n 1 

Linearity (Box R p. 551)

The nth Term Test for Divergence: If a series converges, then lim an  0 . The contrapositive is
n 
true (if lim an  0 , then the series diverges.) The converse is false (if the limit is 0, the series
n 
does not necessarily converge). See table attached.

A geometric series converges to
a1
for 1  r  1 and diverges elsewhere. See table
1 r
attached.
Sections / pages
2
F § 9.4
R § 10.2
Hand-in Homework
R: p.556 – 2, 3, 5, 8, 11, 14, 17, 21, 23, 27, 28, 37, 43, 55
Show work
Due date
Lesson: 3 Class Date ____________________
Learn how to / Learn about: Convergence of Positive Series
Four tests to determine if a positive series converges
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Positive bounded series converge
The Integral Test for convergence See table attached.
Convergence of a p-series See table attached.
Comparison test See table attached.
Limit Comparison Test See table attached.
Sections / pages
3
F § 9.4
R § 10.3
Hand-in Homework
R: p. 566 – 3, 7, 9, 15, 16, 19, 24, 39, 41
49 – 77 odd tell only which test you would try first, you
do not have to work them out unless you want more
practice Show work
Due date
Lesson: 4 Class Date ____________________
Learn how to / Learn about: Series with Negative terms
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Absolutely convergent series
Conditionally convergent series
Absolute convergence implies conditional convergence, but not conversely.
Alternating series test (Leibniz Test). See table attached.
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If N terms are used to approximate a series then the error in the approximation is E  S  S N
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Error bound for alternating series. See table attached.
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Harmonic series

1
1
1
1
 n  1 2  3  4 
(diverges)
n 1


Alternating harmonic series

 1
n 1
n 1
n
Sections / pages
4
F § 9.4
R § 10.5
1 1 1
 1    
2 3 4
(converges to ln(2))
Hand-in Homework
R: p. 574 – 1, 2, 3, 5, 7, 9, 11, 40,
17 – 31 odd – tell only which test you would try first, you
do not have to work them out unless you want more
practice Show work
Due date
Lesson: 5 Class Date ____________________
Learn how to / Learn about: The Ratio Test
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
The Ratio Test See table attached.
The Root Test See table attached.
Sections / pages
5
F § 9.4
R § 10.5
Hand-in Homework
R: p. 578 – 1, 3, 5, 7, 9, 19, 22, 23, 37, 39
43 – 56 all – tell only which test you would try first, you
do not have to work them out unless you want more
practice
Show work
Due date
Summary of the convergence tests that may appear on the AP Calculus BC exam
Test Name
The series …
will converge if
Or will diverge if

a
nth –term test
n 1

Geometric
series
Alternating
series test
 ar
lim an  0
n 
n
n 1
1  r  1
n 1

 (1)n1 an
lim an  0
S  Sn  an1
n 
n 1

 f  x  dx
converges
diverges
p 1
p 1
0  an  bn and
0  bn  an and
1
an  f  n   0

1
n
p-series
n 1
p

Direct
comparison test
a
n 1

 bn converges
n
n 1

a
Ratio test
n 1

 f  x  dx
and
n
|a |
lim n 1  1
n  | a |
n
n
a
1 r
Error bound
n 1
a
Sum =
an1  an and

Integral test
r  1 or r  1
Comments
For divergence
only; the converse
is false.
1
f must be
continuous,
positive and
decreasing.

b
n 1
n
diverges
|a |
lim n 1  1
n  | a |
n
If lim
n 
| an 1 |
1
| an |
the ratio test
cannot be used.
Other useful convergence tests that may be used
Test Name
Limit
comparison
test
The series …

a
n 1
will converge if
Or will diverge if
an  0, bn  0
an  0, bn  0
lim
n
n 
an
L0
bn
 bn converges
n 1

a
n 1
n 

and
Root test
lim
lim an  1
n
n
n 
Comments
an
L0
bn

and
b
n 1
n
diverges
lim an  1
n
n 
The test cannot
be used if
lim n an  1
n 
Sequence Questions
10.
Answers: 10. C, 43%, 61% 22. E 28%, 48% 24. D 42%, 63%