MAC2312 Lesson 3.9 Power Series representation

1
Representation of functions by Power Series
We will use the familiar converging geometric
series form to obtain the power series
representations of some elementary functions.
If iri lt 1, then the power series
The sum of such a converging geometric series is
Conversely, if an elementary function is of the
form
Its power series representation is
By identifying a and r, such functions can be
represented by an appropriate power series.
2
Find a geometric power series for the function
Divide numerator and denominator by 4
Make this 1
a 3/4
r x/4
Use a and r to write the power series.
3
Find a geometric power series for the function
Let us obtain the interval of convergence for
this power series.
f(x)
-4 lt x lt 4
Watch the graph of f(x) and the graph of the
first four terms of the power series.
The convergence of the two on (-4, 4) is obvious.
4
Find a geometric power series centered at c -2
for the function
The power series centered at c is
Divide numerator and denominator by 6
Make this 1
a ½
r c (x 2)/6
f(x)
x -2
5
Find a geometric power series centered at c 2
for the function
Add 4 to compensate for subtracting 4
Since c 2, this x has to become x 2
Divide by 3 to make this 1
Make this to bring it to standard form
6
Find a geometric power series centered at c 2
for the function
This series converges for
x 2
7
Find a geometric power series centered at c 0
for the function
We first obtain the power series for 4/(4 x)
and then replace x by x2
Replace x by x2
Make this to bring it to standard form
f(x)
8
Find a geometric power series centered at c 0
for the function
We obtain power series for 1/(1 x) and 1/(1
x) and combine them.
f(x)
STOP HERE!
9
Find a geometric power series centered at c 0
for the function
We obtain power series for 1/(1 x) and obtain
the second derivative.
Replace n by n 2 to make index of summation n
0
10
Find a geometric power series centered at c 0
for the function
We obtain power series for 1/(1 x) and
integrate it
Then integrate the power series for 1/(1 x )
and combine both.
11
Find a geometric power series centered at c 0
for the function
When x 0, we have
0 0 c
c 0
12
Find a geometric power series centered at c 0
for f(x) ln(x2 1)
We obtain the power series for this and integrate.
f(x)
c 0
When x 0, we have
0 0 c
-1 lt x lt 1
13
Find a power series for the function f(x)
arctan 2x centered at c 0
We obtain the power series for this and integrate.
Replace x by 4x2
14
Find a power series for the function f(x)
arctan 2x centered at c 0
f(x) arctan 2x
-0.5
-0.5
When x 0, arctan 2x 0
0 0 c
c 0