The document uses the ratio test to find the radius of convergence for two power series. For the first series, the ratio test gives an interval of convergence from -2 to 4, so the radius of convergence is 3. For the second series, the ratio test gives the interval of convergence as from -infinity to infinity, so the radius of convergence is infinite.

5. Using the ratio test, we obtain
At x = -2 and x = 4, the corresponding series are, respectively,
These series are convergent alternating series and geometric series, respectively. Therefore
the interval of convergence is (-2, 4), and hence the radius of convergence is  = 3.
 






00 3
1
,
3
1
n
n
n
n
n
 




1 3
1
n
n
n
n
x
 
42-for1,
3
1
1
lim
3
1
)1(31
)1(3
lim 1
1











x
x
n
nx
xn
xn
nnn
nn
n
Find the radius of convergence for the power series below.

6. Using the ratio test, we obtain
Thus the interval of convergence is (-, ), and hence the radius of convergence is
infinite.
   








x
nn
n
x
xn
xn
nn
n
n
-for1,0
!1
!
lim2
)2(!1
)2(!
lim
1
 




1 !
2
n
n
n
x
Find the radius of convergence for the power series below

7. • Finding the general solution of a linear differential equation
depends on determining a fundamental set of solutions of
the homogeneous equation.
• So far, we have a systematic procedure for constructing
fundamental solutions if equation has constant coefficients.
• For a larger class of equations with variable coefficients, we
must search for solutions beyond the familiar elementary
functions of calculus.
• The principal tool we need is the representation of a given
function by a power series.
• Then, similar to the undetermined coefficients method, we
assume the solutions have power series representations, and
then determine the coefficients so as to satisfy the equation.

8. • A power series about the point x0 has the form
and is said to converge at a point x if
exists for that x.
• Note that the series converges for x = x0. It may converge for
all x, or it may converge for some values of x and not others.
 



1
0
n
n
n xxa
 


m
n
n
n
m
xxa
1
0lim

9. • A power series about the point x0
is said to converge absolutely at a point x if the series
converges.
• If a series converges absolutely, then the series also
converges. The converse, however, is not necessarily true.
 



1
0
n
n
n xxa
  





1
0
1
0
n
n
n
n
n
n xxaxxa

10. • One of the most useful tests for the absolute convergence of a
power series
is the ratio test. If an  0, and if, for a fixed value of x,
then the power series converges absolutely at that value of x
if |x - x0|L < 1 and diverges if |x - x0|L > 1. The test is
inconclusive if |x - x0|L = 1.
 



1
0
n
n
n xxa
,lim
)(
)(
lim 0
1
0
0
1
01
Lxx
a
a
xx
xxa
xxa
n
n
nn
n
n
n
n


 



