Power Series
John E. Gilbert, Heather Van Ligten, and Benni Goetz
What’s the payof For introducing all these various tests and applying them to complicated series?
Well, think oF the remarkable series
∞
s
n
= 0
x
n
n
!
= 1 +
x
+
x
2
2!
+
x
3
3!
+
x
4
4!
+
. . .
+
x
n
n
!
+
. . .
=
e
x
we mentioned right at the beginning. OF course, we still have no idea why the sum oF the series is
e
x
;
that’s coming up soon! But at least now we see that the series converges absolutely For all
x
because
by the Ratio test
lim
n
→∞
v
v
v
v
a
n
+1
a
n
v
v
v
v
=
lim
n
v
v
v
v
v
p
x
n
+1
x
n
P
n
!
(
n
+ 1)!
v
v
v
v
v
=
lim
n
x
n
= 0
For each
x
. The series thus has a ±nite sum For each
x
, even iF we haven’t actually determined the sum
oF the series or know why absolute convergence helps.
Now let’s look at Geometric series and Harmonic series in this context. Thinking oF the common
ratio as a variable
x
, not as a number
r
, we get
f
(
x
) =
∞
s
n
= 0
x
n
= 1 +
x
+
x
2
+
x
3
+
x
4
+
. . .
+
x
n
+
. . .
=
1
1

x
(take
a
= 1
For simplicity). The series converges absolutely on
(

1
,
1)
and on this interval its sum is
1
1

x
, so the series provides a
series representation
oF a simple Function From high school days.
But this is calculus, so why not integrate both the Geometric series and its sum:
i
x
0
±
∞
s
n
= 0
t
n
²
dt
=
∞
s
n
= 0
±
i
x
0
t
n
dt
²
=
∞
s
n
= 0
x
n
+1
n
+ 1
=
∞
s
n
= 0
x
n
n
i
x
0
1
1

t
dt
=

ln(1

x
) = ln
³
1
1

x
´
,
assuming that the integral of the inFnite sum is the sum of the inFnite number of integrals
; that’s
coming up too!
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThus
ln
p
1
1

x
P
=
x
+
x
2
2
+
x
3
3
+
x
4
4
+
. . .
+
x
n
n
+
. . . ,
which on letting
x
→
1

becomes
ln(
∞
) = 1 +
1
2
+
1
3
+
1
4
+
. . .
+
1
n
+
. . .
=
∞
.
This provides a nice way of actually linking Geometric series and the divergence of the Harmonic series
that wasn’t at all obvious when we started. There must be a bigger picture  that’s what the Fnal three
lectures on series are about!!
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '11
 Gilbert
 Power Series, Mathematical Series, lim

Click to edit the document details