Math 104 - Calculus I

1
Math 104 - Calculus I

• August 9
• (but first, a quick review)

2
Series of positive terms

• Convergence questions for series of positive
  terms are easiest to understand conceptually.
• Since all the terms a are assumed to be
  positive, the sequence of partial sums S must
  be an increasing sequence.
• So the least upper bound property discussed
  earlier comes into play -- either the sequence of
  partial sums has an upper bound or it doesn't.
• If the sequence of partial sums is bounded above,
  then it must converge and so will the series. If
  not, then the series diverges. That's it.

n
n
3
Tests for convergence of series of positive
terms

• The upper bound observations give rise to
  several "tests" for convergence of series of
  positive terms. They all are based pretty much on
  common sense ways to show that the partial sums
  of the series being tested is bounded are all
  less than those of a series that is known to
  converge (or greater than those of a series that
  is known to diverge). The names of the tests we
  will discuss are...

4
Tests...

• 1. The integral test
• 2. The comparison test
• 3. The ratio test
• 4. The limit comparison test (sometimes called
  the ratio comparison test)
• 5. The root test

TODAY
TODAY
5
The integral test

• Since improper integrals of the form
• provide us with many examples of telescoping
  series whose convergence is readily determined,
  we can use integrals to determine convergence of
  series

6
Integral test cont.

• For example, consider the series
• From the following picture, it is evident that
  the nth partial sum of this series is less than

7
What is the sum?

• The sum of the terms is equal to the sum of the
  areas of the shaded rectangles, and if we start
  integrating at 1 instead of 0, the
• improper integral converges
• (question what is the integral? so what bound
  to you conclude for the series?).
• Since the value of the improper integral (plus 1)
  provides us with an upper bound for all of the
  partial sums, the series must converge.
• It is an interesting question as to exactly what
  the sum is. We will answer it next week.

8
The integral test...
9
Discussion and Connect
Question

• -- for which exponents p does the series
  converge?
• (These are sometimes called p-series, for obvious
  reasons -- these together with the geometric
  series give us lots of useful examples of series
  whose convergence or divergence we know).

10
Error estimates

• Using the picture that proves the integral test
  for convergent series, we can get an estimate on
  how far off we are from the limit of the series
  if we stop adding after N terms for any finite
  value of N.
• If we approximate the convergent series
• by the partial sum
• then the error we commit is less than the value
  of the integral

11
Take a closer look...
12
Question

• A) Converge
• B) Diverge

13
Question
A) Converge B) Diverge
14
Connect
Exercise

• For this latter series, find a bound on the
  error if we use the sum of the first 100 terms to
  approximate the limit. (answer it is less than
  about .015657444)

15
The comparison test

• This convergence test is even more common-
• sensical than the integral test. It says that if
• all the terms of the series are less than
• the corresponding terms of the series
• and if converges, then
• converges also.

16
Reverse

• This test can also be used in reversed -- if
• the b series diverges and the as are bigger
• than the corresponding bs, then
• diverges also.

17
Examples
18
Question
A) Converge B) Diverge
19
Question
A) Converge B) Diverge
20
Convergence Tests...

• 1. The integral test
• 2. The comparison test
• 3. The ratio test
• 4. The limit comparison test (sometimes called
  the ratio comparison test)
• 5. The root test

21
The ratio test

• The ratio test is a specific form of the
  comparison test, where the comparison series is a
  geometric series. We begin with the observation
  that for geometric series, the ratio of
  consecutive terms
• is a constant (we called it r earlier).

22
Ratio test (cont.)

• For other series, even if the ratio of
  consecutive terms is not constant, it might have
  a limit as n goes to infinity. If this is the
  case, and the limit is not equal to 1, then the
  series converges or diverges according to whether
  the geometric series with the same ratio does. In
  other words

23
The ratio test
24
Example
25
Another example
For , the ratio is 1 and the
ratio test is inconclusive. Of course, the
integral test applies to these p-series.
26
Question
A) Converge B) Diverge
27
Question
A) Converge B) Diverge
28
Root test

• The last test for series with positive terms that
  we have to worry about is the root test. This is
  another comparison with the geometric series.
  It's like the ratio test, except that it begins
  with the observation that for geometric series,
  the nth root of the nth term approaches the ratio
  r as n goes to infinity (because the nth term is
  arn and so the nth root of the nth term is
  a1/nr-- which approaches r since the nth root of
  any positive number approaches 1 as n goes to
  infinity.

29
The root test says...
30
Example
31
Question
A) Converge B) Diverge
32
Series whose terms are not all positive

• Now that we have series of positive terms under
  control, we turn to series whose terms can change
  sign.
• Since subtraction tends to provide cancellation
  which should "help" the series converge, we begin
  with the following observation
• A series with and - signs will definitely
  converge if the corresponding series obtained by
  replacing all the - signs by signs converges.

33
Absolutely convergent series

• A series whose series of absolute values
  converges, which is itself then convergent, is
  called an absolutely convergent series.

34
Examples...
Series that are convergent although their series
of absolute values diverge (convergent but not
absolutely convergent) are called conditionally
convergent.
35
Alternating series

• A special case of series whose terms are of both
  signs that arises surprisingly often is that of
  alternating series . These are series whose terms
  alternate in sign. There is a surprisingly simple
  convergence test that works for many of these

36
Alternating series test
37
Example

• The alternating harmonic series clearly
  satisfies the conditions of the test and is
  therefore convergent. The error
• estimate tells us that the sum
• is less than the limit, and within 1/5. Just to
  practice using the jargon, the alternating
  harmonic series, being convergent but not
  absolutely convergent, is an example of a
  conditionally convergent series.

38
Classify each of the following...

• A) Absolutely convergent
• B) Conditionally divergent
• C) Divergent

39
Classify each of the following...

• A) Absolutely convergent
• B) Conditionally divergent
• C) Divergent

40
Classify each of the following...

• A) Absolutely convergent
• B) Conditionally divergent
• C) Divergent

41
Power series

• Last week's project was to try and sum series
  using your calculator or computer. The answers
  correct to ten decimal places are
• Sum((-1)n/(2n1),n0..infinity)
  evalf(sum((-1)n/(2n1),n0.. infinity))
• Sum(1/factorial(n),n0..infinity)evalf(sum(1/fact
  orial(n),n0..infinity))

42
Power series (cont.)

• Sum(1/n2,n1..infinity)evalf(sum(1/n2,n1..infi
  nity))
• Sum((-1)(n1)/n,n1..infinity)evalf(sum((-1)
  (n1)/n,n1..infinity))
• We can recognize these numbers as

43
Two directions

• 1. Given a number, come up with a series that has
  the number as its sum, so we can use it to get
  approximations.
• 2. Develop an extensive vocabulary of "known"
  series, so we can recognize "familiar" series
  more often.

44
Geometric series revisited
45
r as a variable

• Changing our point of view for a minute (or a
  week, or a lifetime), let's think of r as a
  variable. We change its name to x to emphasize
  the point

So the series defines a function (at least for
certain values of x).
46
Watch out...

• We can identify the geometric series when we see
  it, we can calculate the function it represents
  and go back and forth between function values and
  specific series.
• We must be careful, though, to avoid substituting
  values of x that are not allowed, lest we get
  nonsensical statements like

47
Power series

• If you look at the geometric series as a
  function, it
• looks rather like a polynomial, but of infinite
  degree.
• Polynomials are important in mathematics for many
• reasons among which are
• 1. Simplicity -- they are easy to express, to
  add, subtract, multiply, and occasionally divide
• 2. Closure -- they stay polynomials when they are
  added, subtracted and multiplied.
• 3. Calculus -- they stay polynomials when they
  are differentiated or integrated

48
Infinite polynomials

• So, we'll think of power series as "infinite
  polynomials", and write

49
Three (or 4) questions arise...

• 1. Given a function (other than ), can it be
  expressed as a power series? If so, how?
• 2. For what values of x is a power series
  representation valid? (This is a two part
  question -- if we start with a function f(x) and
  form "its" power series, then
• (a) For which values of x does the series
  converge?
• (b) For which values of x does the series
  converge to f(x) ?
• There's also the question of "how fast".

50
continued

• 3. Given a series, can we tell what function it
  came from?
• 4. What is all this good for?
• As it turns out, the questions in order of
  difficulty, are 1, 2(a), 2(b) and 3. So we start
  with question 1

51
The power series of a function of f(x)

• Suppose the function f(x) has the power series

Q. How can we calculate the coefficients a
from a knowledge of f(x)? A. One
at a time -- differentiate and plug in x0!
i
52
Take note...
53
Continuing in this way...
54
Example

• Suppose we know, for the function f, that f(0)1
  and f ' f.
• Then f '' f ', f ''' f '' etc... So f '(0)
  f ''(0) f '''(0) ... 1.
• From the properties of f we know on the one hand
  that So we get that...

55
Good night

• See you Wednesday!