ESSENTIAL CALCULUS CH08 Series

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ESSENTIAL CALCULUSCH08 Series
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In this Chapter

• 8.1 Sequences
• 8.2 Series
• 8.3 The Integral and Comparison Tests
• 8.4 Other Convergence Tests
• 8.5 Power Series
• 8.6 Representing Functions as Power Series
• 8.7 Taylor and Maclaurin Series
• 8.8 Applications of Taylor Polynomials
• Review

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A sequence can be thought of as a list of numbers
written in a definite order The number a1 is
called the first term, a2 is the second term, and
in general an is the nth term.
a1, a2, a3, a4,???,an,???
Chapter 8, 8.1, P412
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NOTATION The sequence a1 ,a2 ,a3 , . . . is
also denoted by
an or
Chapter 8, 8.1, P412
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Chapter 8, 8.1, P413
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Chapter 8, 8.1, P413
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1.DEFINITION A sequence an has the limit L and
we write or
an?L as n?8 if we can make the terms an as
close to L as we like by taking n sufficiently
large. If limn?8an exists, we say the sequence
converges (or is convergent). Otherwise, we say
the sequence diverges (or is divergent).
Chapter 8, 8.1, P414
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Chapter 8, 8.1, P414
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Chapter 8, 8.1, P414
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2.DEFINITION A sequence an has the limit L and
we write or
an?L as n?8 if for every egt0 there is a
corresponding integer N such that
if ngtN then an-Llte
Chapter 8, 8.1, P414
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3. THEOREM If limx?8 f(x)L and f(n)an when n
is an integer, then limn?8 anL.
Chapter 8, 8.1, P415
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5. DEFINITION limn?8 an8 means that for Every
positive number M there is an integer N such
that if ngtN then angtM
Chapter 8, 8.1, P415
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If an and bn are convergent sequences and c
is a constant, then
if
if pgt0 and angt0
Chapter 8, 8.1, P416
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If an bn cn for nn0 and ,then
Chapter 8, 8.1, P416
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6. THEOREM If
Chapter 8, 8.1, P416
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FIGURE7 The sequence bn is squeezed between the
sequences an and cn.
Chapter 8, 8.1, P416
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8. The sequence rn is convergent if -1ltr1 and
divergent for all other values of r.
0 if -1 lt R lt 1 1 if r 1
Chapter 8, 8.1, P418
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9. DEFINITION A sequence an is called
increasing if anltan1 for all n1, that is,
a1lta2lta3lt???, It is called decreasing if angtan1
for all n1. A sequence is monotonic if it is
either increasing or decreasing.
Chapter 8, 8.1, P419
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10. DEFINITION A sequence anis bounded above
if there is a number M such that
an M for all n1 It is bounded below
if there is a number such that
m an for all n1 If it is bounded above
and below, then an is a bounded sequence.
Chapter 8, 8.1, P419
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11.MONOTONIC SEQUENCE THEOREM Every bounded,
monotonic sequence is convergent.
Chapter 8, 8.1, P420
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If we try to add the terms of an infinite
sequence we get an expression of
the Form 1. which is called an infinite series
(or just a series) and is denoted, for short, by
the symbol or
Chapter 8, 8.2, P422
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We consider the partial sums
s1a1 s2a1a2
s3a1a2a3
s4a1a2a3a4 and, in general,

Chapter 8, 8.2, P423
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2. DEFINITION Given a series , let sn denote its
nth partial sum If the sequence sn is
convergent and limn?8 sns exists as a real
number, then the series ?an is called convergent
and we write

or The number s is called the sum of the series.
If the sequence sn is divergent, then the
series is called divergent.
Chapter 8, 8.2, P423
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4. The geometric series is convergent if rlt1
and its sum is Ifr1 , the geometric series
is divergent.
Chapter 8, 8.2, P424
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6.THEOREM If the series is
convergent, then
Chapter 8, 8.2, P426
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The converse of Theorem 6 is not true in
general. If , we cannot
conclude that is convergent.
Chapter 8, 8.2, P427
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7. THE TEST FOR DIVERGENCE If does not
exist or if , then the
series is divergent
Chapter 8, 8.2, P427
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• 8. THEOREM If ?an and ?bn are convergent series,
  then so are the series ?can (where is a
  constant), ?( an bn) , and ? (an - bn) , and
• (ii)
• (iii)

Chapter 8, 8.2, P427
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• THE INTEGRAL TEST Suppose f is a continuous,
  positive, decreasing function on 1,8) and let
  anf(n). Then the series is convergent if
  and only if the improper integral is
  convergent. In other words
• If is convergent, then is
  convergent.
• (b) If is divergent, then is
  divergent.

Chapter 8, 8.3, P433
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1. The p-series is convergent if
pgt1 and divergent if p1.
Chapter 8, 8.3, P434
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• THE COMPARISON TEST Suppose that ?an and ?an are
  series with positive terms.
• If ?bn is convergent and anbn for all n, then
  ?an
• is also convergent.
• (b) If ?bn is divergent and anbn for all n, then
  ?an is also divergent.

Chapter 8, 8.3, P435
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THE LIMIT COMPARISON TEST Suppose that ?an and
?bn are series with positive terms. If where
c is a finite number and cgt0, then either both
series converge or both diverge.
Chapter 8, 8.3, P436
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An alternating series is a series whose terms are
alternately positive and negative. Here are two
examples
Chapter 8, 8.4, P439
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THE ALTERNATING SERIES TEST If the alternating
series satisfies (i)
for all n (ii) then the series is
convergent
Chapter 8, 8.4, P440
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Chapter 8, 8.4, P440
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ALTERNATING SERIES ESTIMATION THEOREM If
is the sum of an alternating series
that satisfies (i) and
(ii) then
The rule does not apply to other types of series.
Chapter 8, 8.4, P442
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DEFINITION A series is called
absolutely convergent if the series of absolute
values is convergent.
Chapter 8, 8.4, P443
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DEFINITION A series is called
conditionally convergent if it is
convergent but not absolutely convergent
Chapter 8, 8.4, P444
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• THEOREM If a series is absolutely
• Convergent, then it is convergent.

Chapter 8, 8.4, P444
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• THE RATIO TEST
• If , then the series
  is
• absolutely convergent (and therefore convergent).
• (ii) If or ,
  then the series
• is divergent.
• (iii) If , the Ratio Test is
  inconclusive
• that is, no conclusion can be drawn about the
• convergence or divergence of .

Chapter 8, 8.4, P445
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• THE ROOT TEST
• If , then the series
  is
• absolutely convergent (and therefore
  convergent).
• (ii) If or
  , then the
• series is divergent.
• (iii) If , the Root Test is
  inconclusive.

Chapter 8, 8.4, P447
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A power series is a series of the form 1. where
x is a variable and the cns are constants called
the coefficients of the series.
Chapter 8, 8.5, P449
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a series of the form 2. is called a power
series in (x-a) or a power series centered at a
or a power series about a.
Chapter 8, 8.5, P449
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3. THEOREM For a given power series there
are only three possibilities (i) The series
converges only when xa. (ii) The series
converges for all x. (iii) There is a positive
number R such that the series converges if
and diverges if
. The number R in case (iii) is called the
radius of convergence of the power series.
Chapter 8, 8.5, P451
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The interval of convergence of a power series is
the interval that consists of all values of x for
which the series converges. In case (i) the
interval consists of just a single point a. In
case (ii) the interval is (-8,8). In case (iii)
note that the inequality x-altR can be rewritten
as a - Rltxlt a R . Thus in case (iii) there are
four possibilities for the interval of
convergence (a-R, aR) (a-R, aR) a-R, aR)
a-R,aR
Chapter 8, 8.5, P451
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Chapter 8, 8.5, P451
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Series Radius of convergence
Interval of convergence
R1
(-1,1)
Geometric series
R0
Example 1
0
2,4)
R1
Example 2
(-8,8)
Example 3
R8
Chapter 8, 8.5, P452
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xlt0
Chapter 8, 8.6, P454
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2. THEOREM If the power series
has radius of convergence Rgt0, then the function
f defined by is differentiable (and therefore
continuous) on the interval (a - R, a R)
and (i) (ii) The radii of convergence of the
power series in Equations (i) and (ii) are both R.
Chapter 8, 8.6, P456
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NOTE 1 Equations (i) and (ii) in Theorem 2 can be
rewritten in the form (iii) (iv)
Chapter 8, 8.6, P456
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5.THEOREM If f has a power series representation
(expansion) at a, that is, if
x-altR then its
coefficients are given by the formula
Chapter 8, 8.7, P461
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6.
The series in Equation 6 is called the Taylor
series of the function f at a (or about a or
centered at a).
Chapter 8, 8.7, P461
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7. For the special case a0 the Taylor series
becomes
This case arises frequently enough that it is
given the special name Maclaurin series.
Chapter 8, 8.7, P461
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In the case of the Taylor series, the partial
sums are Notice that Tn is a polynomial n
of degree n called the nth-degree Taylor
polynomial of f at a.
Chapter 8, 8.7, P462
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If we let so
that then Rn(x) is called the remainder of the
Taylor series.
Chapter 8, 8.7, P463
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8. THEOREM If f(x)Tn (x)Rn (x) , where Tn is
the nth-degree Taylor polynomial of f at a
and for x-altR , then f is equal to the sum
of its Taylor series on the interval x-altR.
Chapter 8, 8.7, P463
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9. TAYLORS FORMULA If f has n1 derivatives in
an interval I that contains the number a, then
for x in I there is a number z strictly between x
and a such that the remainder term in the Taylor
series can be expressed as
Chapter 8, 8.7, P463
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for every real number x
Chapter 8, 8.7, P463
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for all x
Chapter 8, 8.7, P464
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Chapter 8, 8.7, P464
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for all x
Chapter 8, 8.7, P465
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for all x
Chapter 8, 8.7, P466
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The Maclaurin series of f(x)(1x)k is This
series is called the binomial series.
Chapter 8, 8.7, P466
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The traditional notation for the coefficients in
the binomial series is and these numbers are
called the binomial coefficients.
Chapter 8, 8.7, P467
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18.THE BINOMIAL SERIES If k is any real number
and xlt1, then
Chapter 8, 8.7, P467
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R1
R8
R8
R8
R8
R8
Chapter 8, 8.7, P468