15A Objective:
Sequences
AP Calculus:
Big Idea 4: Series
Resources:
Lesson:





Rogawski, 2012,
EU 4.1: LO 4.1A / EK 4.1A1
2nd
ed: 10.1 p537-548
Stewart, 1999, 4th ed: 12.1 p727-737
SEQUENCE: {a1, a2, a3, …} can be denoted by {an} or
;
an = general term, a1 = 1st term, a1 = preceding term
Two types of formulas to generate a sequence: EXPLICIT or RECURSIVE
The sequence CONVERGES to a limit L If
Otherwise the sequence DIVERGES.
If the terms increase without bound, we say the sequence diverges to infinity
SEQUENCE {an} DEFINED BY A FUNCTION f(x):
If
when n is an integer, then
Note: the only real difference between sequence {an} and function f( n ) converging/diverging
is than n has to be an integer
Types of Sequences
o ARITHMETIC SEQUENCE example: {1, 3, 5, 7, 9, 11, ...}
recursive: a1 = c; an = an-1 + d when n > 2
explicit: an = a1 + (n-1) d
will be divergent unless d = 0
o GEOMETRIC SEQUENCE {rn}, example: {1, 2, 4, 8, 16, ...}
recursive: g1 = c; gn = gn-1 r when n > 2
explicit: gn = g1(r)n-1
o FIBONACCI SEQUENCE {fn} = {1, 1, 2, 3, 5, 8, 13, 21, ...}
recursive: f1 = 1, f2 = 1, fn = fn-1 + fn-2 when n > 3;
is divergent
o HARMONIC SEQUENCE


explicit:
is convergent to 0
Know the Limit Laws of Sequences and Squeeze Theorem of Sequences on p541
For all n > 1, the sequence {an} is…
 INCREASING if an < an+1 for all n
 DECREASING if an > an+1 for all n
 It is MONOTONIC if it is either increasing or decreasing
o …and given M is a number, the sequence {an} is
 BOUNDED ABOVE if an < M
 BOUNDED BELOW if an > M
 It is a BOUNDED SEQUENCE if bounded above and below
o Every bounded, monotonic sequence is convergent
15A Objective: Sequences
(continued)
 TI CALCULATIOR
o GRAPHING SEQUENCES
You can use the sequential graphing mode to do both explicit and recursive functions
Otherwise use parametric equations, where xt = t and yt = {explicit formula}
o TI LIST OPERATIONS: (2nd STAT)
OPS/seq(formula/list, index, start, finish [, optional increment]):
will generate sequence of terms
example: seq(x^2,x,1,3) = {1 4 9 16}
Problems:
[15A1]
10.1 p545: 3, 5, 7, 15, 17, 27, 35, 37, 67
(Stewart, 1999 4th ed) 10.1 p545 (1-14, 15-37 odd, 39-46 graph & guess, 47-51 odd, 52-59)
15B Objective:
Summing an Infinite Series
AP Calculus:
Big Idea 4: Series
EU 4.1: LO 4.1A / EK 4.1A1, EK 4.1A2, EK 4.1A3
LO 4.1B / EK 4.1B1
Resources:
Rogawski, 2012, 2nd ed: 10.2 p548-559
Lesson:
 SERIES is a sum of a sequence which can be…
o FINITE
Stewart, 1999, 4th ed: 12.2 p738-747

INFINITE SERIES:
or
(index n may start at any integral value k)
PARTIAL SUMS {sn} is a sequence of sums of an increasing number of terms in a finite series

example:
would product the partial sums of
CONVERGENCE OF AN INFINTE SERIES
o
Given an infinite series
for the first four terms
,
let sn denote its nth partial sum where

If the sequence {sn} converges to limit s, then the series
and s is known as the sum. Otherwise series is divergent.
Types of Series:
converges where
GEOMETRIC SERIES:
GEOMETRIC SERIES TEST:
Any rational number (repeating decimal) can be written as an infinite geometric series

HARMONIC SERIES:
is divergent
Note: the harmonic sequence converges to 0, but the series is divergent
TELESCOPING SERIES
is a series where terms cancel out in pairs and the sum collapses into just two terms
example by partial fraction decomposition:

Review Sigma Rules on p551
o
15B Objective:

Summing an Infinite Series (continued)
THEOREM: If
converges, then
However the converse is not true. Just because
that
converges, then
The closest thing would be the…

does not guarantee
Harmonic Series is a good example.
TEST FOR DIVERGENCE:
If
or does not exist, then
is divergent
 TI CALCULATOR
o LIST/MATH/sum(sequence/list) would be your sigma-notation series
or MATH/summation (explicit formula, index, start, finish) can be used with TI-84
MathPrint example: sum(seq(x^2,x,1,3)) = 30
o LIST/MATH/cumSum(sequence/list) to view partial sums
example: cumSum(seq(x^2,x,1,3)) = {1 5 14 30}
Problems:
[15B1]
10.2 p556: 1, 3, 5, 9, 11, 15, 17, 21, 23, 27, 31, 37, 41, 45
(Stewart, 1999 4th ed) 12.2 p745 (3-8 use cumSum and graphing, 9-10, 11-45 odd, 52)
15C Objective:
Convergence of Series with Positive Terms
AP Calculus:
Big Idea 4: Series
EU 4.1: LO 4.1A / EK 4.1A3
2nd
Resources:
Rogawski, 2012,
ed: 10.3 p559-569
Stewart, 1999, 4th ed: 12.3, 12.4
Lesson:
 DICHOTOMY THEOREM:
A positive series S converges if its partial sums SN (which form an increasing sequences)
remains bounded. Otherwise it diverges.

INTEGRAL TEST (goes back to improper integrals)
Given f is continuous, positive, decreasing function on [1,∞), an = f( n )

if
converges, then
if
diverges, then
converges
diverges
p-SERIES TEST (also back to improper integrals)
converges if p > 1, diverges if p < 1

COMPARISON TEST: Given

LIMIT COMPARISON TEST: Given
and
are series with positive terms
If
converges and an < bn for all n, then
converges
If
diverges and an > bn for all n, then
diverges
Note: although the test says for all n, you only need to verify that n > N where N is a fixed
integer
and
are series with positive terms
If
where c > 0 and is a finite number, then both series converge or diverge
Use similar fractions like p-series if possible
Problems:
[15C1]
10.3 p566: 1-31 every other odd, 39, 43, 47
**Mixed Review of Methods: 49-78

(Stewart, 1999 4th ed) 12.3 p754 (3-27 odd); 12.4 p759 (1-2, 3-31 odd, 38)
15D Objective:
Absolute and Conditional Convergence
AP Calculus:
Big Idea 4: Series
Resources:
Lesson:
Rogawski, 2012, 2nd ed: 10.4 p569-575
EU 4.1: LO 4.1A / EK 4.1A4, EK 4.1A5, EK 4.1A6
LO 4.1B / EK 4.1B2, EK 4.1B3
Stewart, 1999, 4th ed: 12.5, 12.6

A series
is ABSOLUTELY CONVERGENT if
converges
Theorem: Any absolutely convergent series converges


A series
is CONDITIONALLY CONVERGENT if it converges but
diverges
ALTERNATING SERIES: series whose terms alternate positively and negatively
Will involve (-1)n or (-1)n-1 as part of its explicit formula

ALTERNATING SERIES TEST:
(Leibniz Test for Alternating Series)
To show
converges, you must show all three conditions...
(i) bn > 0
(ii) bn > bn+1 for all n (> N); use algebra and/or derivatives
(iii)
If the series doesn't meet all requirements, try other tests (ex. Test for Divergence)

ALTERNATING SERIES REMAINDER (ERROR):
If
converges, the size of the remainder/error is smaller than bn+1 - the
absolute value of the first neglected term; |L - sn| < bn+1 where sn is the partial sum of n terms
Problems:
[15D1]
10.4 p574: 1-9 odd, 34
**Mixed Review of Methods: 17-32
(Stewart, 1999 4th ed) ALT SERIES: 12.5 p764 (3-19 odd, 21-26 show convergence and remainder/error)
15E Objective:
Ratio and Root Tests
AP Calculus:
Big Idea 4: Series
Resources:
Lesson:

Rogawski, 2012,
EU 4.1: LO 4.1A / EK 4.1A4, EK 4.1A5, EK 4.1A6
2nd
ed: 10.5 p575-579
Stewart, 1999, 4th ed: 12.5, 12.6
RATIO TEST: Given
L < 1, then
absolutely converges, thus converges
L > 1 or DNE, then
diverges
L = 1, then tells you nothing and you'll need another method

ROOT TEST: Given
L < 1, then
absolutely converges, thus converges
L > 1 or DNE, then
diverges
L = 1, then tells you nothing and you'll need another method
 Beware of REARRANGEMENTS - where you rearrange the order of terms
If
absolutely converges with sum s, then any rearrangement of
has the same sum s.
If
conditionally converges, rearrangements of
can have different sums. In fact, there
is a rearrangement for all possible sums.
Problems:
[15E1] 10.5 p578:
1-17 odd, 57 (Proof of Root Test) **Mixed Review of Methods: 43-56
(Stewart, 1999 4th ed)
ABS/COND CONV, RATIO & ROOT TEST: 12.6 p770 (1-35 odd)
Mixed Review: 12.7 p754 (1-38)