Calculus Section 9.9 Geometric Power Series
-Find a geometric power series that represents a function.
-Construct a power series using series operations.
Homework: page 674 #’s 5, 8, 9, 10, 11, 12
The final type of series expansion is based off of the geometric series.
Review of geometric series: write a geometric series for a1 = 2 and r = 3.
Write a geometric series for a1 = 2 and r = -3.
Write a geometric series for a1 = 1 and r = x.
For a geometric series to converge, the radius for the series must be _________________. If we borrow the
a
summation formula for a convergent geometric series, sum  1 , we can write a power series for some
1 r
types of functions.
Examples) Find a Power Series and Interval of Convergence for Each Function
1
1
f ( x) 
f ( x) 
c=0
c=2
1 2x
4 x
f ( x) 
3
,c=1
3  2x
Random Examples of Power/Taylor/Maclaurin Series)
x
h(x)
h’(x)
h’’(x)
h’’’(x)
1
10
5
6
-2
2
6
4
-7
8
3
-3
-2
9/4
3
Write a second-degree Taylor Polynomial for f about x = 2, and use it to approximate f(2.3). If the third
derivative of f satisfies the inequality f ''' ( x)  9 for all x in the interval, use the Lagrange error bound to find an
interval [a,b] such that a ≤ f(2.3) ≤ b.
If the first 4 terms of the Taylor expansion for f(x) about x = 0 are 15  4 x 
3 2
x  2 x 3 , then what is f '''(0) ?
2