Math 104 - Calculus I August 9 (but first, a quick review…)
Dec 20, 2015
Math 104 - Calculus I
August 9
(but first, a quick review…)
Series of positive termsConvergence 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.
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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...
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
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:
1dx)x(f
For example, consider the series
From the following picture, it is evident that the nth partial sum of this series is less than
Integral test cont.
1
12
nn
n
x1 dx
1 21
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.
1 2 dxx1
The integral test...
11
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.dx
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)x(
x
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Discussion and Connect
-- 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).
1
1
nn p
Question…
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
1
)(fn
n
N
nN ns
1
)(f
Ndxx)(f
Take a closer look...
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Question
A) Converge
B) Diverge
diverge?or converge series theDoes1
1 2
nnn
Question
A) Converge
B) Diverge
diverge?or converge series theDoes1
1
)arctan(2
nn
n
Connect
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)
Exercise
The comparison testThis 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.
1nna
1nnb
1nna
1nnb
Reverse
This test can also be used in reversed -- if
the b series diverges and the a’s are bigger
than the corresponding b’s, then
diverges also.
1nna
Examples:
diverges.
converges.
1)sin(
12
1
nnn
n
nnn
Question
diverge?or converge series theDoes5
21
kk
A) Converge
B) Diverge
Question
diverge?or converge series theDoes1
21
2
nnn
A) Converge
B) Diverge
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
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). n
n
a
a 1
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:
The ratio test:
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exist),not doeslimit theif(or 1r If
1r if -diverges-
1r if -converges-
series then the, lim If1
1
n
nn
n
nar
a
a
Example:
converges. series theso
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1 lim
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have we,!
1For
1
1
nn
n
a
a
n
nnn
n
n
n
For , the ratio is 1 and the ratio
test is inconclusive.
Of course, the integral test applies to these p-series.
Another example:
1
1
npn
Question
diverge?or converge 5
! series theDoes
1
nn
n
A) Converge
B) Diverge
Question
diverge?or converge ln
series theDoes2
kke
k
A) Converge
B) Diverge
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.
The root test says...
1.r ifdrawn becan conclusion no and
1r if diverges series the--
1r if converges series the--
then,lim ifthat
1
1
1
nn
nn
nn
a
a
ra n
Example
root test. by the converges
521
n
n
n
n
Question
diverge?or converge 1 series theDoes1
n
nne
A) Converge
B) Diverge
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.
Absolutely convergent series
A series whose series of absolute values converges, which is itself then convergent, is called an absolutely convergent series.
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Examples...
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Series that are convergent although their series of absolute values diverge (convergent but not absolutely convergent) are called conditionally convergent.
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:
Alternating series test:
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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.
...1 41
31
21
127
41
31
211
Classify each of the following...
A) Absolutely convergent
B) Conditionally divergent
C) Divergent
2 ln
)1(
n
n
nn
Classify each of the following...
A) Absolutely convergent
B) Conditionally divergent
C) Divergent
13
sin
k k
k
Classify each of the following...
A) Absolutely convergent
B) Conditionally divergent
C) Divergent
12 5
cos
n n
nn
Power seriesLast 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/(2*n+1),n=0..infinity) = evalf(sum((-1)^n/(2*n+1),n=0..
infinity));
Sum(1/factorial(n),n=0..infinity)=evalf(sum(1/
factorial(n),n=0..infinity));
0
12)1( 7853981635.
nn
n
0
!1 718281828.2
nn
Power series (cont.)Sum(1/n^2,n=1..infinity)=evalf(sum(1/n^2,n=1..infinity));
Sum((-1)^(n+1)/n,n=1..infinity)=evalf(sum((-1)^
(n+1)/n,n=1..infinity));
We can recognize these numbers as
1
1 6644934068.12
nn
1
)1( 6931471806.)1(
nn
n
).2(ln and , , 64
2 e
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.
Geometric series revisited
1). (provided -1
...
:series geometric the
friend, oldour begin with We
432
0
rr
a
ararararaarn
n
r as a variableChanging 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).
)1(for 1
...)(f0
32
xx
aaxaxaxaaxx
n
n
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
!!1...168421...22221 432
Power seriesIf 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
Infinite polynomials
So, we'll think of power series as "infinite polynomials", and write
0
33
2210 ...
n
nn xaxaxaaxa
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".]
x
ax
1)(f
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:
The power series of a function of f(x)
Suppose the function f(x) has the power series...)(f 4
43
32
210 xaxaxaxaax
Q. How can we calculate the coefficients a from a knowledge of f(x)?
A. One at a time -- differentiate and plug in x=0!
i
Take note...
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a
xaxaxaax
a
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a
aaaaa
Continuing in this way...
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zero,at evaluated f of derivativenth
:generalIn
etc.... (0)/24,f (0)/6,f 43
na
aa
ExampleSuppose 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...
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!...
!4!3!21
0
432
n
nx
n
xxxxxe
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Good night…
See you Wednesday!