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Material Type: Notes; Class: Calculus II; Subject: Mathematics; University: Pellissippi State Technical Community College; Term: Unknown 1989;
Typology: Study notes
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Power Series
A power series is an infinite series of the form
n 0
c n
x
n
c 0
c 1
x c 2
x
2
c 3
x
3
c n
x
n
where x is a variable is called a power series. More generally, series of the form
n 0
c n
x − a
n
c 0
c 1
x − a c 2
x − a
2
c n
x − a
n
is called a power series centered at a or the power series about a, where a is a constant.
Note: to simplify notation we let
x − a
0
1 even when x a. For a power series centered at
a there are only THREE possibilities:
( 1 ) The series converges only when x a.
( 2 ) The series converges for all x.
( 3 ) There is a positive real number R such that the series converges if |x − a |
R and
diverges if |x − a| R.
The number R is called the radius of convergence of the power series. In case (1), R 0.
In case (2), R . The set of all values of x for which the power series converges is called the
interval of convergence of the power series.
We use the Ratio Test to find the radius of convergence of a power series. For a power
series centered about a with a finite radius of convergence R 0, we must determine whether
or not the series converges at the endpoints of the interval of convergence (i.e. at a − R &
a R).
Find the interval of convergence for the following power series.
n 0
n!x
n
n 0
n
x
2 n 1
2 n 1 !
n 1
x
n
n
n 0
n
x 1
n
n
n 1
x − 2
n
n
2
Term - by - term differentiation and integration of functions defined by power series :
n 0
c n
x − a
n
have radius of convergence R 0. Then the function
f
x
n 0
c
n
x − a
n
c
0
c
1
x − a c
2
x − a
2
c
3
x − a
3
is differentiable (and therefore continuous) on the interval a − R, a R and
( a ) f
′
n 0
nc n
x − a
n− 1
c 1
2 c 2
x − a 3 c 3
x − a
2
n 0
c n
x − a
n 1
n 1
C c 0
x − a c 1
x − a
2
c 2
x − a
3
The series in (a) and (b) have radius of convergence R. Convergence or divergence at the
endpoints of the interval of convergence may differ from the original power series.
Find f
′
n 1
x
n
n
n 1
x
n
n!
find g
′
x. Do you recognize this function?