Recall that the Taylor polynomial of degree
n for a differentiable function
at x = c is
If we let n approach infinity, we arrive at the Taylor Series for f(x)
at x = c.
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
(z - c)n+1
(n + 1)!
then the Taylor Series converges.
Find the McLaurin Series expansion for
f(x) = cos(x)
We construct the following table.
Hence we have the series
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
sin(x) centered at x = p/2
sinh(x) centered at x = 0
The Standard Normal Distribution function is defined by
We define the probability as follows:
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
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
1 - cos x =
- + ...
Now divide both sides by x to get
1 - cos x
- + ...
When x = 0, the right hand side becomes zero,
hence so does the left hand side.
Prove L'Hopital's Rule using power series.
Addition and Subtraction of Power
We have that the power series representation of
- x) +
1 - x
Find the power Series Representation for
+ arctanh x
Multiplication of Power Series
Suppose we have two power series
What is the power series for
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
1 + x +
x4 + ...
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 =
The second degree term gives us
1 + c2 = 0. Hence c2 =
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
x - x2 +
x3 + ...
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