Math 104 - Calculus I

1
Math 104 - Calculus I

• Part 8
• Power series, Taylor series

2
Series

• First .. a review of what we have done so far
• 1. We examined series of constants and learned
  that we can say everything there is to say about
  geometric and telescoping series.
• 2. We developed tests for convergence of series
  of constants.
• 3. We considered power series, derived formulas
  and other tricks for finding them, and know them
  for a few functions.
• 4. We used the ratio tests to determine
  intervals on which power series converge, and use
  the other tests to check convergence at the
  endpoints of the intervals.

3
Geometric series

• a/(1-r) a ar ar2 ar3 ...
• provided rlt 1.
• We often use partial fractions to detect
  telescoping series, for which we can calculate
  explicitly the partial sums sn

4
Tests for convergence for series of constants

• Fundamental divergence test (nth term must go to
  zero for convergence to be possible)
• Integral test
• Comparison and limit comparison tests
• Ratio test
• Root test
• Alternating series test

5
Power series

• f(x) a0 a1 x a2 x2 a3 x3 ...
• where an f(n)(0)/n!
• We know the series for ex, sin(x), cos(x),
• 1/(1-x), and a few related functions.

6
Convergence of power series

• Before we get too excited about finding series,
  let's make sure that, at the very least, the
  series converge.
• Next week, we'll deal with the question of
  whether they converge to the function we expect.
  But for now, we'll assume that if they converge,
  they converge to the function they "came from".
• (Strictly speaking, this is not always true --
  but it is true for a large class of functions,
  which includes nearly all the ones encountered in
  basic science and mathematics. This fact was not
  fully appreciated until the early part of the
  twentieth century.)
• Fortunately, most of the question of whether
  power series converge is answered fairly directly
  by the ratio test.

7
Recall that...

• for a series of constants , we have that
  the series converges (absolutely) if the the
  limit of the absolute value of is less than one,
  diverges if
• the limit is greater than one, and the test is
  indeterminate if the limit equals one.
• To use the ratio test on power series, just leave
  the x there and calculate the limit for each
  value of x. This will give an inequality that x
  must satisfy in order for the series to converge.
  

8
For the series for the exponential function...
9
Your turn...

• Calculate the series for the function sin(x) and
  determine for which x the series converges.

10
Heres a more interesting example
11
What remains...
12
Final Conclusion
13
OK, your turn...
A. -1 lt x lt 1 B. -2 lt x lt 2 C. 1/2 lt x lt 1/2
D. -2 lt x lt 2 E. -1/2 lt x lt 1/2
14
One more...
A. -1 lt x lt 1 B. -1 lt x lt 1 C. 1 lt x lt 1 D. -1
lt x lt 1 E. 0 lt x lt 1
15
From these examples,

• ...it should be apparent that power series
  converge for values of x in an interval that is
  centered at zero, i.e., an interval of the form
  -a, a , (-a, a, -a, a) or (-a, a) (where a
  might be either zero or infinity). The interval
  is called the interval of convergence and the
  number a is called the radius of convergence .

16
Lets go back
To finding series of functions
17
The other way
18
Try this...

• Take the derivative of the series for sin(x) to
  get

19
Integrate both sides of the geometric series from
0 to x to get
20
Negate both sides and replace x by (-x)
everywhere to get
21
Start from the geometric series again...
2
And substitute x for x everywhere it appears to
get
22
A challenge to think about...

• How to get the other one from previously

23
Application of Series

• 1. Limits Series give a good idea of the
  behavior of functions in the neighborhood of 0
• We know for other reasons that
• We could do this by series

24
This can be used on complicated limits...

• Calculate the limitA. 0
• B. 1/6
• C. 1
• D. 1/12
• E. does not exist

25
Application of series (continued)

• 2. Approximate evaluation of integrals Many
  integrals that cannot be evaluated in closed form
  (i.e., for which no elementary anti-derivative
  exists) can be approximated using series (and we
  can even estimate how far off the approximations
  are).
• Example Calculate to the nearest 0.001.

26
We begin by...
27
According to Maple...

• The last series is an alternating series with
  decreasing terms. We need to find the first one
  that is less than 0.0005 to ensure that the error
  will be less than 0.001. According to Maple
• evalf(1/(7factorial(3))), evalf(1/(9factorial(4)
  )),evalf( 1/(11factorial(5)))
• evalf(1/(13factorial(6)))

.02380952381, .004629629630, .0007575757576
.0001068376068
28
Keep going...

• So it's enough to go out to the 5! term. We do
  this as follows
• Sum((-1)n/((2n1)factorial(n)),n0..5)
  sum((-1)n/((2n1) factorial(n)),n0..5)
• evalf()

.7467291967.7467291967
29
and finally...

• So we get that to the nearest
  thousandth.
• Again, according to Maple, the actual answer (to
  10 places) is
• evalf(int(exp(-x2),x0..1))

.74669241330
30
Try this...

• Sum the first four nonzero terms to approximate
• A. 0.7635
• B. 0.5637
• C. 0.3567
• D. 0.6357
• E. 0.6735

31
Application of series (cont.)

• 3. Differential equations Another important
  application of series is to find "solutions" to
  differential equations (when other methods fail).
  
• Earlier, we found the series for based on
  the differential equation it satisfies (y ' y
  ), together with the initial condition y(0)1.

32
Airy functions

• The equation y '' xy 0 has no "elementary"
  solution (the solutions of this equation are
  called "Airy functions", named after a famous
  British Royal Astronomer). But we can find series
  for the solution that satisfies y(0)1, y '(0)0,
  as follows
• From the initial conditions, we can assume that
  the series for y(x) begins

33
Airy (continued)

• Then the series for series

34
So...
We work our way from left to right -- since y
xy is supposed to be zero, every coefficient must
be zero. Therefore
35
So the series for y(x) begins as follows...
36
What is the denominator of the x term of this
series?
9

• A. 720
• B. 1080
• C. 1440
• D. 4440
• E. 12960

37
Application of series (cont.)

• 4. Algebraic equations depending on a parameter
  ("regular perturbations").
• Consider the equation . We could calculate
  the solutions of this equation using the
  quadratic formula, but it will be instructive to
  think of the two solutions (x) as functions of
  the parameter a .
• We will find the power series of one of these
  functions.
• Keep in mind that we are thinking of x as a
  function of a (and not the other way around)!
• Write x f(a).
• First, if a0, there are two roots, x 2 and x
  -2. We'll calculate the series for the larger
  root, so f(0)2.

38
Work it out...
39
Right to left
40
Graph

• To see how good the approximation is, we plot the
  "actual" solution (in blue) and our approximation
  (in red) on the same graph

41
And.

• What is the beginning of the series for the other
  root?

42
A max/min problem

• During World War II, the blood of thousands of
  recruits had to be tested for various diseases.
  The incidence of the diseases was quite low (less
  than one per hundred). Assume that the
  probability that a random person has the disease
  is p (for some p between 0 and 1, but much closer
  to 0 than to 1). Let q 1 - p be the probability
  that the person does not have the disease.
• Because the tests were costly, someone had the
  bright idea to pool several (x) samples at a
  time. If the pooled sample tested negative, then
  all x samples were negative. But if the pooled
  sample tested positive, then each individual
  sample must be tested again.

43
max/min
44
max/min
45
Rewrite
46
Undaunted...
we try expanding the terms in powers of p. We
know that ln(1-p) -p ... and If we
substitute this much into the equation, we get
The solution of this is (which is at
least consistent with the limit!). What we have
here is an example of an asymptotic series , or,
a series of powers other than whole numbers. We
can calculate more coefficients by expanding the
equation in higher powers of p.