Chapter 7 Infinite Sequences and Series

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Chapter 7 Infinite Sequences and Series
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7.1 Sequences

• A sequence is a list of numbers a1, a2, a3, ,
  an, in a given order.
• Each of a1, a2, a3 and so on represents a
  number. These are the terms of the
• sequence.
• The integer n is called the index of an, and
  indicates where an occurs in the list.
• Order is important.
• We can think of sequence a1, a2, a3, , anas a
  function that sends 1 to a1, 2 to a2,
• 3 to a3, and in general sends the positive
  integer n to the nth term an.

For example, the function associated with the
sequence 2, 4, 6, 8, , 2n, sends 1 to a12, 2
to a24, and so on. The general behavior of the
sequence is described by the formula an2n.
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Remarks

• We can equally well make the domain the integers
  larger than a given number n0 (may gt1), and we
  allow the sequence of this type also.
• 2. Sequences can be described by writing rules
  that specify their terms, such as
• or by listing terms,

We also sometimes write
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Graphically Represent Sequences
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Convergence and Divergence
Sometimes the numbers in a sequence approach a
single value as the index n Increases. For
example, an1/n whose terms approach 0 as n gets
large, or an1-1/n approach 1 as n gets
large. On the other hand, some sequences like
an(-1)n1n bounce back and forth Between 1 and
-1, never converging to a single value.
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This definition is very similar to the
Definition of the limit of a function f(x) as x
tends to ?
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Examples

• Example
• Example

any constant k.
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Diverges to Infinity
The sequence also diverges, because
The sequence 1, 02, 3, -4, 5, -6, 7, -8, and
1, 0, 2, 0, 3, 0, are examples of such
divergence.
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Calculating Limits of Sequences
Since sequences are functions with domain
restricted to the positive integers, we have
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Remark

• Theorem 1 does not say that, for example, that if
  the sum an bn
• has a limit, then each of the sequences an
  and bn have limits.
• One consequence of Theorem 1 is that every
  nonzero multiple of a
• divergent sequence an diverges.

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Examples
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Sandwich Theorem for Sequences
An immediate consequence of Theorem 2 is that ,
if bn?cn and cn?0, then bn ? 0 because cn ? bn
? cn, we use this fact in the next example.
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Examples
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Continuous Function Theorem for Sequences
Example Show that
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Example

• Example Show that

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Using LHopitals Rule
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Commonly Occuring Limits
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Examples
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Recursive Definitions

• So far, we have calculated each an directly from
  the value of n. But sequences are often defined
  recursively by giving
• The values (s) of the initial term or terms, and
• A rule, called a recursion formula, for
  calculating any later term from terms that
  precede it.
• Example
• The statement a11, and anan-11 define the
  sequences of positive integers.
• The statement a11, a21, and an1anan-1
  defines the sequence of Fibonacci numbers.

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Nondecreasing Sequences

• Example The following sequences are
  nondecreasing
• The sequence 1, 2, 3, , n, of natural numbers
• The sequence ½, 2/3, ¾, , n/(n1),
• The constant sequence 3.

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Bounded Nondecreasing Sequences
Here are two kinds of nondecreasing
sequencesthose whose terms increase beyond any
finite bound and those whose terms do not.

• Examples
• The sequence 1, 2, 3, , n has no upper bound.
• The sequence ½, 2/3, 34, , n/(n1), , is
  bounded above by M1. 1 is also
• the least upper bound.

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A nondecreasing sequence that is bounded from
above always has a least upper Bound. We also
prove that if L is the least upper bound then the
sequence converges to L.
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The Nondecreasing Sequenc Theorem.
Theorem 6 implies that a nondecreasing sequence
converges when it is bounded from above. It
diverges to infinity if it is non bounded from
above. The analogous results hold for
nonincreasing sequences.
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7.2 Infinite Series
An infinite series is the sum of an infinite
sequence of numbers. a1a2a3an The goal of
this section is to understand the meaning of such
an infinite sum and to develop methods to
calculate it. The sum of the first n terms sn
a1a2a3an is an ordinary finite sum, is
called The nth partial sum. As n gets larger, we
expect the nth partial sums to get closer and
closer to a limiting value in the same sense that
the terms of a sequence approach a limit.
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Example
For example, to assign meaning to an expression
like 11/21/41/81/16 We add the terms one at
a time from the beginning and look for a pattern
in how These partial sums grow.
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The partial sums form a sequence whose nth term
is The sequence of partial sums converges to 2
because . We
say the sum of the infinite series
11/21/41/2n-1 is 2.
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Note it is convenient to use sigma notation to
write the series.
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Geometric Series
Geometric series are series of the form In which
a and r are fixed real numbers and a?0. The
series can also be written as . The
ratio r can be positive, or negative.
The formula a/(1-r) for the sum of a geometric
series applies only when the summation index
begins with n1 in ( or with n0 in
).
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Examples
Example The series
converges.
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Examples

• Example Find the sum of the series

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Divergent Series
One reason that a series may fail to converge is
that its terms dont become small. Example The
series diverges because the partial
sums eventually outgrow Every preassigned number.
Each term is greater than 1, so the sum of n
terms is greater than n.
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Examples
Example
Example The series 11/21/21/41/41/41/4,
1/2n1/2n1/2n This diverges, however, the
terms of the series form a sequence that
converges to 0.
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Combining Series
Note ?(anbn) can converge when ?an and ?bn both
diverge.
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Examples
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Remarks

• We can add a finite number of terms to a series
  or delete a finite number of terms without
  altering the series convergence or divergence,
  although in the case of the convergence this will
  usually change the sum.
• As long as we preserve the order of its term, we
  can reindex any series without altering its
  convergence.

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7.3 The Integral Test

• Given a series, we want to know whether it
  converges or not. In this section and the next
  two, we study series with nonnegative terms.
• Since the partial sums from a nondecreasing
  sequence, the Nondecreasing Sequence Theorem tell
  us the following

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Example

• The series
  is called the harmonic series.
• The harmonic series is divergent, but this
  doesnt follow from the nthTerm Test.
• The reason it diverges is because there is no
  upper bound for its partial sums.

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Example
Example Does the following series converge?
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The Integral Test
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Error Estimation
If a series ?an is shown to be convergent by the
integral test, we may want to estimate the size
of the remainder Rn between the total sum S of
the series and its nth partial sum sn.
If we add the partial sum sn to each side of the
inequality in (1), we get
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7.4 Comparison Tests
We have seen how to determine the convergence of
geometric series, p-series, and a few others. We
can test the convergence of many more series by
comparing Their terms to those of a series whose
convergence is known.
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The Limit Comparison Test
We now introduce a comparison test that is
particularly useful for series in which an is a
ration function of n.
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Example Cont.
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7.5 The Ratio and Root Tests
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The Root Test
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7.6 Alternating Series, Absolute and Conditional
Convergence
A series in which the terms are alternately
positive and negative is an alternating Series.
For example

• Series (1), called the alternating harmonic
  series, converges.
• Series (2), a geometric series with ratio r-1/2,
  converges.
• Series (3) diverges because the nth term does not
  approach zero.

(1)
(2)
(3)
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Alternating Series Test
Example The alternating harmonic series
converges by Theorem 14.
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Graphical Interpretation of the Partial Sums
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The Alternating Series Estimation Theorem
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Example
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Absolute and Conditional Convergence
The geometric series in Example 2 converges
absolutely, while the alternating harmonic
series does not converge absolutely.
The alternating harmonic series converges
conditionally.
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The Absolute Convergence Test
Absolute convergence is important for two
reasons. First, we have good tests for
convergence of series of positive terms. Second,
if a series converges absolutely, then it
converges, as we now prove.
Note we can rephrase Theorem 16 to say that
every absolutely convergent series Converges.
However, the converse statement is false Many
convergent series do not converge absolutely
(such as the alternating harmonic series).
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Examples
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Examples
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Rearranging Series
If we rearrange the terms of a conditionally
convergent series, we get different Results. In
fact, it can be proved that for any real number
r, a given conditionally Convergent series can
be rearranged so its sum is equal to r. So we
must always Add the terms of a conditionally
convergent series in the order given.
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Summary
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7.7 Power Series
Now that we can test infinite series for
convergence, we can study sums that look Like
infinite polynomials. We call these sums poewr
series because they are defined As infinite
series of powers of some variable, in our case x.
Equation (1) is the special case obtained by
taking a0 in Equation (2).
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Example
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We now think of the partial sums of the series on
the right as polynomials Pn(x) that approximate
the function on the left.
The figure left shows the graph of f(x), and the
approximating polynomials ynPn(x) for n0, 1, 2,
and 8.
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Example
The power series
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The convergence Theorem for Power Series
The next result shows that if a power series
converges at more than one value, Then it
converges over an entire interval of values.
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Corollary
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The Term-by-Term Differentiation Theorem
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Examples
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The Term-by-Term Integration Theorem
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Examples
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Examples
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The Series Multiplication Theorem for Power Series
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7.8 Taylor and Maclaurin Series

• This section shows how functions that are
  infinitely differentiable generate power series
  called Taylor series.
• Q If a function f(x) has derivatives of all
  orders on an interval I, can it be expressed as a
  power series on I? And if it can, what will its
  coefficients be?

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The Maclaurin series generated by f is often just
called the Taylor series of f.
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Examples

• Find the Taylor series generated by f(x)1/x at
  a2. Where, if anywhere, does the series converge
  to 1/x?

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Taylor Polynomials
The higher-order Taylor polynomials provide the
best polynomial approximations of their
respective degrees.
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Examples

• Find the Taylor series and the Taylor polynomials
  generated by f(x)ex at x0.

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7.10 The Bionomial Series
This section introduces the binomial series for
estimating powers and roots of binomial Expression
s (1x)m.
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Examples

• Let m-1, the binomial series formula gives the
  familiar geometric series
• (1x)-1 1 x x2 - x3 (-1)kxk

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Examples