Telescoping series

In mathematics, a telescoping series is a series whose general term




t

n




{\displaystyle t_{n}}
can be written as




t

n


=

a

n


−

a

n
+
1




{\displaystyle t_{n}=a_{n}-a_{n+1}}
, i.e. the difference of two consecutive terms of a sequence



(

a

n


)


{\displaystyle (a_{n})}
.As a consequence the partial sums only consists of two terms of



(

a

n


)


{\displaystyle (a_{n})}
after cancellation. The cancellation technique, with part of each term cancelling with part of the next term, is known as the method of differences.
For example, the series





∑

n
=
1


∞




1

n
(
n
+
1
)





{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n(n+1)}}}
(the series of reciprocals of pronic numbers) simplifies as









∑

n
=
1


∞




1

n
(
n
+
1
)









=

∑

n
=
1


∞



(



1
n


−


1

n
+
1




)














=

lim

N
→
∞



∑

n
=
1


N



(



1
n


−


1

n
+
1




)














=

lim

N
→
∞



[


(

1
−


1
2



)

+

(



1
2


−


1
3



)

+
⋯
+

(



1
N


−


1

N
+
1




)


]














=

lim

N
→
∞



[

1
+

(

−


1
2


+


1
2



)

+

(

−


1
3


+


1
3



)

+
⋯
+

(

−


1
N


+


1
N



)

−


1

N
+
1




]














=

lim

N
→
∞



[

1
−


1

N
+
1




]

=
1.






{\displaystyle {\begin{aligned}\sum _{n=1}^{\infty }{\frac {1}{n(n+1)}}&{}=\sum _{n=1}^{\infty }\left({\frac {1}{n}}-{\frac {1}{n+1}}\right)\\{}&{}=\lim _{N\to \infty }\sum _{n=1}^{N}\left({\frac {1}{n}}-{\frac {1}{n+1}}\right)\\{}&{}=\lim _{N\to \infty }\left\lbrack {\left(1-{\frac {1}{2}}\right)+\left({\frac {1}{2}}-{\frac {1}{3}}\right)+\cdots +\left({\frac {1}{N}}-{\frac {1}{N+1}}\right)}\right\rbrack \\{}&{}=\lim _{N\to \infty }\left\lbrack {1+\left(-{\frac {1}{2}}+{\frac {1}{2}}\right)+\left(-{\frac {1}{3}}+{\frac {1}{3}}\right)+\cdots +\left(-{\frac {1}{N}}+{\frac {1}{N}}\right)-{\frac {1}{N+1}}}\right\rbrack \\{}&{}=\lim _{N\to \infty }\left\lbrack {1-{\frac {1}{N+1}}}\right\rbrack =1.\end{aligned}}}

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