Creating Power Series From Functions
The Geometric Power Series
Recall that
Substituting x for r, we have
We write
Milking the Geometric Power Series
By using substitution, we can obtain power series expansions
from the geometric series. Example 1
Substituting x^{2} for x, we have
Example 2
Multiplying by x we have
Example 3
Suppose we want to find the power series
for
1
f(x) =
2x  3
centered at x =
4. We rewrite the function as
1
1
=
2(x  4) + 8 
3 2(x  4) + 5
Example 4
Substituting x for x, we have
Example 5
Substituting x^{2} in for x in
the previous example, we have
Example 6
Taking the integral of the previous
example, we have
Exercise Find the power series that represents the
following functions:

ln(1 + x)

tanh^{1}x

(1  x)^{2}
Integrating Impossible Functions
We can use power series to integrate functions where there are no
standard techniques of integration available.
Example:
Use power series to
find the integral
Then use this integral to approximate
Solution:
Notice that this is a very difficult integral to solve. We resort to power series. First we use the
series expansion from Example 6, replacing x with
x^{2}.
Integrating we arrive at the solution
Now to solve the definite integral, notice that when we plug in 0 we get
0,
hence the definite integral is
Using the first 5 terms to approximate this we get 0.300
Notice that the error is less than the next term (which comes from x^{23}/253)
E < 1/253 = .004.
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