# Taylor Series (4/9/04)

```Clicker Question 1

What is the interval of convergence of

( x   )n

n
n 1
– A. (-, )
– B. (-  1, - + 1)
– C. (  1,  + 1)
– D. [-  1, - + 1)
– E. [-  1, - + 1]
Clicker Question 2

What is the interval of convergence of

n
n
!
x

n 0
–
–
–
–
–
A. (-, )
B. converges nowhere
C. (-n  1, n + 1)
D. (-1, 1)
E. {0}
Taylor Series (4/21/14)
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Given a nice smooth function f (x):
What is the best constant function to
approximate it near 0?
Best linear function to approximate it near 0?
Best quadratic function to approximate it
near 0?
Best cubic function to approximate it near 0?
Etc. etc.
Taylor Series continued

If f(x) is a function for which we can compute
all of its derivatives (i.e., first derivative f '(x) ,
second derivative f (2)(x), third derivative
f (3)(x), and so on), then we can write down a
representation of f as a power series in terms
of the values of those derivatives at the
center of the interval of convergence. This is
called the Taylor Series for f(x).
Taylor Series Centered at 0
(also known as Maclaurin Series)

We get that, on the interval of convergence,
f(x) = f(0) + f '(0) x + f (2)(0) x2 / 2! +…
+ f (n)(0) xn / n! +…

=

f ( n ) (0) x n / n!
n 0

To estimate f , we can use the first so many
terms (as a calculator does).
Examples of Series Centered at 0
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Since if f (x) = ex , then for every n,
f (n)(x) = ex also, so f (n)(0) = 1. Hence the
Taylor Series for ex is very simple:
ex = 1 + x + x2/2! + … + xn/n! + …,
which converges everywhere.
Show that the Taylor series for sin(x) is
sin(x) = x – x3/3! + x5/5! – x7/7! + ….
By computing the derivatives of sin at 0.
Taylor Series Centered at a
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If it is easier (or more useful) to evaluate f ‘s
derivatives at a point a rather than zero, then we
simply center the power series at a and evaluate the
derivative at a:
f(x) = f(a) + f '(a) (x-a) + f (2)(a) (x-a)2 / 2!
+ …+ f (n)(a) (x-a)n / n! +…
For example, what is the Taylor Series for ln(x)
centered at x = 1?
What is its interval of convergence?
Assignment for Wednesday
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