Transcript Calculus for the Natural Sciences

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
an 1
an
is a constant (we called it r earlier).
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:

an 1
If lim
 r , then theseries  an
n  a
n 1
n
- -convergesif r  1
- -divergesif r  1
If r  1 (or if thelimit does not exist),
then no conclusioncan be drawn.
Example:

1
For  , we have
n 1 n!
an 1
n!
1
lim
 lim
 lim
 0,
n  a
n  ( n  1)!
n  n  1
n
so theseries converges.
Another example:

1
For  p , the ratio is 1 and the ratio
n 1 n
test is inconclusive.
Of course, the integral test applies to these
p-series.
Question

n!
Does theseries  n convergeor diverge?
n 1 5
A) Converge
B) Diverge
Question

ln k
Does theseries  k convergeor diverge?
k 2 e
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...
thatif lim an
 1n 
n 
 r , then

- - theseries  an convergesif r  1
n 1

- - theseries  an divergesif r  1
n 1
and no conclusioncan be drawn if r  1.
Example

n
 n 



n 1  2n  5 
convergesby theroot test.
Question


Does theseries  1  e
n 1
A) Converge
B) Diverge

n n
convergeor 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
Another way tosay thisis to thinkof the termsan

of theseries  an as being positiveand negative,
n 1
and then theseries obtainedby changingall minus

signs to plus would be  an - the" series of absolute
n 1
values."
A series whose series of absolute values converges,
which is itself then convergent, is called an
absolutely convergent series.

Examples...

( 1 )n

( 1 )n  1  1  1  1  1  1  ... is divergent (since an doesn' t
n 1

n2
1  1  ... is absolutely convergent .
 1  14  19  16
25
n 0
tend to zero) and (or course) its series of absolute values is divergent.


n 1
( 1 )( n 1 )
1  1  1  1  1  ... is convergent (as we will see)

1

n
2 3 4 5 6
even though its series of absolute values is divergent.
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:

Let  (1) n an  a0  a1  a2  a3  ... be an alternating series.
n 0
If (theabsolute valuesof) the termsare decreasing, thatis
a0  a1  a2  ...
and if lim an  0, then theseries converges. Moreover,thedifference
n 
between the limit of theseries and thepartialsum sn has thesame sign
as and is less in absolute value that thefirst omittedterm(an 1 ).
Example:
1
1
1
1



The alternating harmonic series
2
3
4  ...
clearly satisfies the conditions of the test
and is therefore convergent. The error
estimate tells us that the sum 1  12  13  14  127
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.
Classify each of the following...
(1)

n  2 n ln n

n
A) Absolutely convergent
B) Conditionally convergent
C) Divergent
Classify each of the following...

sin k

3
k 1 k
A) Absolutely convergent
B) Conditionally convergent
C) Divergent
Classify each of the following...
n cos n

2
n 1 n  5

A) Absolutely convergent
B) Conditionally convergent
C) Divergent
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/(2*n+1),n=0..infinity) = evalf(sum((-1)^n/(2*n+1),n=0..
infinity));


n 0
( 1) n
2 n 1
 .7853981635
Sum(1/factorial(n),n=0..infinity)=evalf(sum(1/factorial(n),n=0..infinity)
);


n 0
1
n!
 2.718281828
Power series (cont.)
Sum(1/n^2,n=1..infinity)=evalf(sum(1/n^2,n=1..infinity));


n 1
1
n2
 1.6644934068
Sum((-1)^(n+1)/n,n=1..infinity)=evalf(sum((-1)^
(n+1)/n,n=1..infinity));


n 1
( 1) ( n1)
n
 .6931471806

We can recognize these numbers as 4 , e,
2
6
and ln(2).
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
We begin with our old friend,
thegeometricseries :

 ar
n
 a  ar  ar  ar  ar  ... 
2
n 0
a
(providedr  1).
1- r
3
4
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:

a
f ( x)   ax  a  ax  ax  ax  ... 
(for x  1)
1 x
n 0
n
2
3
So the series defines a function (at least for
certain values of x).
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  2  2  2  2  ...  1  2  4  8  16  ...  1!!
2
3
4
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
Infinite polynomials
So, we'll think of power series as "infinite
polynomials", and write

a x
n 0
n
n
 a0  a1 x  a2 x  a3 x  ...
2
3
Three (or 4) questions arise...
a
1. Given a function (other than f ( x)  1  x ), 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".]
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 ( x)  a0  a1x  a2 x  a3 x  a4 x  ...
2
3
4
Q. How can we calculate the coefficients ai from a
knowledge of f(x)?
A. One at a time -- differentiate and plug in x=0!
Take note...
First (or zerot h): f (0)  a0  a1 0  a2 0 2  a3 03  ...  a0
so we have t hata0  f(0)
Second (first?): It seems reasonablet o writ e
f ( x)  a1  2a2 x  3a3 x 2  4a4 x 3  ..., so we should have
a1  f (0)
Anot herderivative:
f ( x)  2a2  6a3 x  12a4 x 2  20a5 x 3  ...,
so we should havea2  f (0)/2
Continuing in this way...
a3  f (0)/6,a4  f (0)/24,etc....
In general:
an  nth derivativeof f evaluatedat zero,
divided by n!
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
x
f
(
x
)

e
that
So we get that...
2
3
4
n

x
x
x
x
e x  1  x    ...  
2! 3! 4!
n  0 n!
(Set x  1 to get one of thesums we did previously!)