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Math 104 - Calculus I
Part VII:
More tests for convergence;
Power series
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:
drawn. becan conclusion nothen
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
,01
1 lim
)!1(
! lim lim
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.
values."
absolute of series" the- be wouldplus tosigns
minus all changingby obtained series then theand
negative, and positive being as series theof
terms theof think tois say this y toAnother wa
1
1
nn
nn
n
a
a
a
Examples...
divergent. is valuesabsolute of series its thougheven
see) will we(as convergent is 1
divergent. is valuesabsolute of series its course)(or and zero) totend
tdoesn' (sincedivergent is 1111111
.convergent absolutely is 1
61
51
41
31
21
1n
1
0n
251
161
91
41
1
1
1
2
...
a...)(
...
n)(
nn
nn
)(
)n(
n
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:
).( termomittedfirst that the valueabsolutein less is and as
sign same thehas sum partial theand series theoflimit ebetween th
difference theMoreover, converges. series then the,0lim if and
...
is that ,decreasing are terms theof) valuesabsolute (the If
series. galternatinan be ...)1(Let
1
210
32100
n
n
nn
nn
n
a
s
a
aaa
aaaaa
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 convergent
C) Divergent
2 ln
)1(
n
n
nn
Classify each of the following...
A) Absolutely convergent
B) Conditionally convergent
C) Divergent
13
sin
k k
k
Classify each of the following...
A) Absolutely convergent
B) Conditionally convergent
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...
(0)/2f have should weso
...,201262)(f
:derivativeAnother
(0)f
have should weso ,...432)(f
write toreasonable seemsIt :(first?) Second
f(0) that have weso
...000)0(f :zeroth)(or First
2
35
2432
1
34
2321
0
03
32
210
a
xaxaxaax
a
xaxaxaax
a
aaaaa
Continuing in this way...
n!by divided
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...
)!previously did wesums theof oneget to1(Set x
!...
!4!3!21
0
432
n
nx
n
xxxxxe
xex )(f
