11.2 PROPERTIES OF POWER
SERIES
Math 6B
Calculus II
POWER SERIES
 Power
Series centered at x = 0.
 Power Series centered at x = a.
THE FUNCTION 1/(1 – X)

The function 1/(1 – x) can be rewritten into a
power series.
1
1 x

 1  x  x  x  ... 
2
3

k 0
x
k
x 1
HOW TO TEST A POWER SERIES FOR
CONVERGENCE
 Use
the Ratio test (or Root test) to find the
interval where the series converges
absolutely. Ordinarily this is an open
interval x  a  R or a – R < x < a + R
 If the interval of absolute convergence is
finite, test for convergence or divergence
at each endpoint. Use an appropriate
test.
HOW TO TEST A POWER SERIES FOR
CONVERGENCE
 If
the interval of absolute
convergence is a – R < x < a + R, the
series diverges for x  a  R
THE RADIUS AND INTERVAL OF
CONVERGENCE
Possible behavior of  c k ( x  a )
 There is a positive number R (also known
as the radius of convergence) such that the
series diverges for all x  a  R but
converges for x  a  R . The series may
or may not converge at either endpoint
x = a – R and x = a + R.
k
THE RADIUS AND INTERVAL OF
CONVERGENCE
 The
series converges absolutely for
every x
R
 
 The
series converges at x = a and
diverges everywhere else. (R = 0 )
COMBINING POWER SERIES
Suppose the power series 𝑐𝑘 𝑥𝑘 and
𝑑𝑘 𝑥𝑘 converge absolute to 𝑓(𝑥) and 𝑔(𝑥)
respectively, on an interval I.
 Sum and difference: The power series
𝑐𝑘 𝑥𝑘 ± 𝑑𝑘 𝑥𝑘 converges absolutely to
𝑓(𝑥) ± 𝑔(𝑥) on I.
 Multiplication by a power: The power
series 𝑥 𝑚 𝑐𝑘 𝑥 𝑘 = 𝑐𝑘 𝑥 𝑘+𝑚 converges
absolutely to 𝑥 𝑚 𝑓(𝑥) on I, provided m is
an integer such that 𝑘 + 𝑚 ≥ 0 for all
terms of the series
COMBINING POWER SERIES
If ℎ 𝑥 = 𝑏𝑥 𝑚 , where m is
a positive integer and b is a real number,
the power series 𝑐𝑘 (ℎ 𝑥 )𝑘 converges
absolutely to the composite function
𝑓 ℎ 𝑥 , for all x such that h(x) is in I.
 Composition:
DIFFERENTIATION AND INTEGRATION OF A
POWER SERIES
If the pow er series

c k ( x  a ) has radius of con vergence
k
R  0, then the function f defined by

f ( x )  c 0  c1 ( x  a )  c 2 ( x  a )  ... 
2

ck ( x  a )
k
k 0
is differentiable (and therefore continu ous) on the interval
( a  R , a  R ) and
i

f '( x )  c1  2 c 2 ( x  a )  3 c 3 ( x  a )  ... 
2

k 1
kc k ( x  a )
k 1
DIFFERENTIATION AND INTEGRATION OF A
POWER SERIES
 ii  
f ( x ) dx  C  c 0 ( x  a )  c1

C 
c
k 0
(x  a)
n
(x  a)
2
2
 c2
(x  a)
3
k 1
k 1
The radii of convergence of the power
series in Equations (i) and (ii) are both R.
3
 ...