Taylor Series

Taylor Series

Recall that the Taylor polynomial of degree n for a differentiable function f(x) centered
at x = c is If we let n approach infinity, we arrive at the Taylor Series for f(x) centered at x = c.

 Definition    The Taylor Series for f(x) centered at x = c is If c = 0 we call this series the Mclaurin Series for f(x).  Recall that the error of the nth degree Taylor Polynomial is given by

f (n+1)(z)
R =                        (z - c)n+1
(n + 1)!

Hence if then the Taylor Series converges.

Example

Find the McLaurin Series expansion for

f(x) = cos(x)

Solution

We construct the following table.
 n f (n)(x) f (n)(0) 0 cos x 1 1 -sin x 0 2 -cos x -1 3 sin x 0 4 cos x 1 5 -sin x 0 6 -cos x -1

Hence we have the series

x2        x4        x6        x8
1  -           +         -         +           - ...
2!        4!        6!        8!

Notice that the series only contains even powers of x and even factorials.  Even numbers can be represented by 2n.  Also notice that this is an alternating series, hence the McLaurin series is Exercises  Find the Taylor series expansion for

1. sin(x) centered at x = p/2

2. sinh(x) centered at x = 0

Statistics

The Standard Normal Distribution function is defined by

 Normal Distribution Function We define the probability as follows:

 Definition of Probability Example:

Use McLaurin series and the fact that to approximate the probability of getting a "B" in this class if the average is 70 and the standard deviation is 10 and the instructor grades on a "curve".  A "B" corresponds to between 1 and 2 standard deviations from the mean, hence we need to compute We can calculate the first many terms on the calculator to get an approximate value of

0.76

In the first quarter you learned a proof that In the second quarter you used L'Hopitals rule.  Now we will do it a third way:  We have Hence

x2         x3
1  -  cos x  =                     + ...
2          3

Now divide both sides by x to get

1  -  cos x          x          x2
=                    + ...
x                 2          3

When x = 0, the right hand side becomes zero, hence so does the left hand side.

Exercise

Prove L'Hopital's Rule using power series.

Addition and Subtraction of Power Series

 Theorem Suppose that we have two functions and their power series representations and Then  Example:

We have that the power series representation of

1
ln(1 - x) +
1 - x

is Exercise

Find the power Series Representation for

arctan x + arctanh x

Multiplication of Power Series

Suppose we have two power series and What is the power series for

f(x)g(x)

Consider the following example.  Let We can multiply these series as though they were finite series.  We collect the coefficients:

• The constant term is 1.

• The first degree term is 1 + 1 = 2.

• The second degree term is 1 + 1 + 1/2 = 5/2.

• The third degree term is 1 + 1 + 1/2 + 1/6 = 8/3

• The fourth degree term is 1 + 1 + 1/2 + 1/6 + 1/24 = 65/24

We can continue this process indefinitely, or better yet use a computer to generate the terms.

The series is

5             8              65
1 + x +       x2   +        x3  +           x4  + ...
2             3              24

Division of Power Series

Suppose we want to find the power series representation of We multiply by the denominator and equate coefficients:

(c0 + c1x + c2x2 + ...)(1 + x + x2/2 + x3/6 + x4/24 + ...) = (x - x3/3 +  x5/5- x7/7 +...)

• The constant coefficient gives us  c0 = 0.

• The first degree term gives us c0 + c1 = 1. Hence c1 = 1.

• The second degree term gives us 1 + c2 =  0. Hence c2 = -1.

• The third degree term gives us 1/2 - 1 + c3 = -1/3.  Hence c3 = 1 - 1/2 - 1/3 = 1/6.

and so on.

The series is

1
x - x2 +       x3 + ...
6