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Power Series
Power Series—Introduction

A power series is a series of the form n= 0cnxn = c0 + c1x + c2x2 + c3x3 + ...
Example 1:

Say cn = 1 for all n, we have n= 0xn as our first power series. So?
Well,
what we really want to know is: for what values of x will

n converge?

x
n=0
So, start asking yourself this question:
Can x be 0, 1/2 , –1/2, 3/4, –3/4, 1, –1, 3/2, –3/2, 2, –2, and so on?
Yes, when x is any
of

these numbers, n= 0xn
becomes convergent
geometric series
since |r| < 1
Conclusion:

No, when x is any
of

these numbers, n= 0xn
becomes divergent
geometric series
since |r|  1
n converges when x is any number between –1 and 1, i.e., –1 < x < 1,

x
n=0
or the open interval: (–1, 1).
Power Series—Interval and Radius of Convergence

Recall our power series  xn converges for x  (–1, 1).
n=0
Therefore, we say (–1, 1) is the interval of convergence (IOC) and we
define the radius of convergence (ROC) as half of the length of IOC.
Since the length of the interval
Radius of Convergence = 1
(i.e., the distance from –1 to 1)
is 2, therefore, the ROC is 1.
–1
0
1
Example 2:
Interval of Convergence = (–1, 1)

n

n!x
n=0
Again, ask yourself this question:
Can x be 0, 1/2 , –1/2, 3/4, –3/4, 1, –1, 3/2, –3/2, 2, –2, and so on?
Yes, if x is 0, it
will make every
term 0
No, if x is any of these
numbers, n= 0n!xn will diverge
ROC = 0
–1
0
1
IOC = {0}
Power Series—The Three Types of IOCs
Example 3:

n/n!

x
n=0
Once more, ask yourself this question:
Can x be 0, 1/2 , –1/2, 3/4, –3/4, 1, –1, 3/2, –3/2, 2, –2, and so on?

Yes, if x is any of these numbers, n= 0xn/n! will converge
Radius of Convergence = 
–2
–1
0
1
2
Interval of Convergence = (–, )
The Three Types of IOCs:

For any given power seriesn= 0cnxn, there are only three possibilities:
(1) The series converges on an interval with a finite length (Example 1)
(2) The series converges at only one number (Example 2)
(3) The series converges for all real numbers (Example 3)
Power Series—More on IOCs and ROCs
Examples:
IOC
ROC
1)
[1, 4)
½(3) = 1.5
(–2, 2]
½(4) = 2
{½}
0
2)
3)
0
1
2
3
4
–2 –1
0
1
2
3
–2 –1
0
1
2
3
Power Series—How to Guess the IOC?
Example 1:
 ( x  3) n

n
n 1
One number, in particular, will obviously make the series converge.
What is this number? [Hint: this number will make every term = 0.]
x=3
Of course, x may be other numbers too. If so, on the number line,
we begin at 3 and move to the right and left to obtain our interval of
convergence.




0
1
2
If x = 2, then the series
(1)
which
n
n 1

becomes 
n
is a convergent series
Therefore, the IOC is [2, 4), and ROC = 1.
3
4
If x = 4, then the series
 1
becomes  which is
n 1n
a divergent series
Power Series—How to Guess the IOC? (cont’d)
Example 2:
 ( x  1) n
 2
n
n 1
What is the number that makes every term = 0?
x=1
Again, on the number line, we begin at 1 and move to the right and
left to obtain our interval of convergence.




0
If x = 0, then the series
1
2
3
4
If x = 2, then the series
(1)
becomes  2 which
n 1 n
becomes 
is a convergent series
a convergent series

n

1
2 which is
n 1 n
Therefore, the IOC is [0, 2], and ROC = 1.
Power Series—How to Find the IOC in General?
Answer: Kind of like doing the Ratio Test
( x  3)n
Example 1: 
n
n 1
( x  3) n
an 
n

x–3
and
a
 x  3
a
1
Step 1: Find n 1 : n1  an1  
an
an
an
n 1
Step 2: Find lim
n 
n 1

( x  3) n1
an1 
n 1
n( x  3)
n

n 1
( x  3) n
n( x  3)
n
an 1
: lim
 lim
 x  3  1 x  3  x  3
n
n

an
n 1
n 1
|x – 3| < 1
–1 < x – 3 < 1
+3
+ 3 +3
2 < x <4
Step 4: Check whether endpoints work or not:
Step 3: Set above limit < 1 and solve for x:
(1) n
If x = 2, the series becomes 
which is a convergent series, so 2 is included.
n
n 1


1
which is a divergent series, so 4 should be excluded.
n 1n
If x = 4, the series becomes 
IOC is [2, 4), and ROC = 1.
Power Series—How to Find the IOC in General? (cont’d)
Recap:
an1
1) We find lim
and set it < 1, and solve for x. (Steps 1–3)
n a
n
2) We then check the endpoints (to see whether they will make the
power series converges or not) by substituting each endpoint into
the series. (Step 4)
Example 2:
 (2 x  1) n

n2
n 1
Example 3:
 ( 1) n (2 x) 2 n

n!
n 0
Example 4:
 n( x  2) n
 n1
3
n 0