Name__________________________________
Period ________

1) Geometric Series:
x

n
Calculus
Lesson 9.1 Day 2
1  x  x 2  ...  x n  ...
n 0
Sigma
Series

1
1 x
Sum Formula
Find the value(s) of x for which the given geometric series converges. Find the sum of the series.
Power Series are series of algebraic functions. We use power series to approximate transcendental functions.
Power Series

An expression of the form
c x
n 0
n
n
 c0  c1 x  c2 x 2  ...  cn x n  ... is a power series centered at x  0 .

An expression of the form
 c  x  a
n 0
n
n
 c0  c1  x  a   c2  x  a   ...  cn  x  a   ... is a power series
2
centered at x  a . The term cn  x  a  is the nth term; the number a is the center.
n
2.) Find a power series that represents
1
on (-1,1).
1 x
n
3.) Find a power series that represents
1
on (-1/2,1/2).
1  2x
4.) Find a power series that represents
x
on (-1,1).
1 x
5.) Find a power series that represents
1
and give its interval of convergence.
5 x
6.) Find a power series that represents
1
and give its interval of convergence.
3x
Term-By-Term Differentiation
1
 1  x  x 2  x 3  x 4  ...  x n  ... on the interval (-1,1), find a power series to represent
7.) Given that
1 x
1
.
2
1  x 
Term-By-Term Integration.
1
n
 1  x  x 2  x3  x 4  x5  ...    x   ... on the interval (-1,1), find a power series to
8.) Given that
1 x
represent ln(1  x) .
We have been looking at graphs of power series. We can also use power series to evaluate expressions.
ln(2)  0.6931471806
Let x  1 and complete the table.
Number of
Terms
1
Expression
Approximation
x
Numerical
Approximation
1
ln(2)  approximation
2
x - x2/2
0.5
0.1931471806
3
x - x2/2 + x3/3
0.8333
0.1401861527
4
x - x2/2 + x3/3 - x4/4
0.58333
0.1098138472
5
6
0.3068528194
Additional Questions:
Find the interval of convergence and the function of x represented by the geometric series.
9.)
3 x
n
n 0
 x 1 
10.)  5 

n 0  2 


n
64.) Use differentiation to find a series for f ( x) 
n

11.)
n
n 0
2
1  x 
 cos
3
. (Use the work in #7 to help.)
Challenge Question:
x 2 x3 x 4
xn
If f  x   1  x     ...   ... , what well-known function do you suppose f to be?
2! 3! 4!
n!
x