Alternating Series
John E. Gilbert, Heather Van Ligten, and Benni Goetz
When some terms of an infinite series take positive values, while others take negative values, the
convergence is often improved. For instance, when
(i)
1 +
1
2
+
1
3
+
1
4
+
...
+
1
n
+
... ,
(ii)
1

1
2
+
1
3

1
4
+
...
+ (

1)
n
1
n
+
... ,
the first series diverges, yet the second series converges because adding and subtracting provides
cancellation
. Let’s get an idea why: grouping the terms in pairs gives
parenleftbigg
1

1
2
parenrightbigg
+
parenleftbigg
1
3

1
4
parenrightbigg
+
...
+
parenleftbigg
1
2
n

1
2
n
+ 1
parenrightbigg
+
...
=
1
1
.
2
+
1
3
.
4
+
...
+
1
2
n
(2
n
+ 1)
+
... .
But
summationdisplay
n
1
n
2
converges, while
lim
n
→ ∞
n
2
parenleftBigg
1
2
n
(2
n
+ 1)
parenrightBigg
=
1
4
, so by the Limit Comparison test,
1 +
1
2
2
+
...
+
1
n
2
+
...
converges
=
⇒
1
1
.
2
+
1
3
.
4
+
...
+
1
2
n
(2
n
+ 1)
+
...
converges
.
This can easily be turned into a proof showing that series (ii) converges.
Series (ii) is called the
Alternating Harmonic Series
. Its structure and properties lead to
Definition:
a series is said to be an
Alternating Series
when
it has the form
summationdisplay
n
(

1)
n
−
1
a
n
with
a
n
>
0
,
i.e., when it can
be written
a
1

a
2
+
a
3

a
4
+
...
+ (

1)
n
a
n
+
... ,
a
n
>
0
,
or, in other words, when
consecutive terms alternate
in sign.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
There’s a very useful test for determining whether an alternating series converges:
Alternating Series Test:
an infinite series
summationdisplay
n
(

1)
n
−
1
a
n
with
a
n
>
0
converges if
•
a
n
≥
a
n
+1
for all
n
≥
N
,
•
lim
n
→ ∞
a
n
= 0
.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '11
 Gilbert
 Mathematical Series, lim

Click to edit the document details