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Question 0510
Sigma Notation
2010 Paper 2 Question 2 Variant
Question
Generate new
(a)
Prove by the method of differences that
∑
r
=
3
n
1
(
r
−
1
)
(
r
+
1
)
=
5
12
−
1
2
n
−
1
2
n
+
2
.
\sum_{r=3}^n \frac{1}{(r - 1) (r + 1)} = \frac{5}{12} - \frac{ 1 }{ 2 n } - \frac{ 1 }{ 2 n + 2 }.
r
=
3
∑
n
(
r
−
1
)
(
r
+
1
)
1
=
12
5
−
2
n
1
−
2
n
+
2
1
.
[4]
(b)
Explain why
∑
r
=
3
∞
1
(
r
−
1
)
(
r
+
1
)
{\displaystyle \sum_{r=3}^\infty \frac{1}{(r - 1) (r + 1)}}
r
=
3
∑
∞
(
r
−
1
)
(
r
+
1
)
1
is a convergent series, and state the value of the sum to infinity.
[2]
Answer
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