Telescoping Series
We illustrate the idea behind the Telescoping Series using the following series as our example:
¥
å
n = 1
1
n(n+1)
Each term can be decomposed into partial fractions:
1
n(n+1)
=
1
n
-
1
n+1
.
The m-th partial sum can therefore be written as follows.
S
m
=
m
å
n = 1
1
n(n+1)
=
m
å
n = 1
æ
ç
è
1
n
-
1
n+1
ö
÷
ø
=
æ
ç
è
1
1
-
1
2
ö
÷
ø
+
æ
ç
è
1
2
-
1
3
ö
÷
ø
+
æ
ç
è
1
3
-
1
4
ö
÷
ø
+...+
æ
ç
è
1
m-1
-
1
m
ö
÷
ø
+
æ
ç
è
1
m
-
1
m+1
ö
÷
ø
After cancellation, we obtain
S
m
= 1-
1
m+1
Since
lim
m
®
¥
S
m
= 1
, the series converges and has a sum 1.