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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
=
2
n
1
r
(
r
+
2
)
=
5
12
−
1
2
n
+
2
−
1
2
n
+
4
.
\sum_{r=2}^n \frac{1}{r (r + 2)} = \frac{5}{12} - \frac{ 1 }{ 2 n + 2 } - \frac{ 1 }{ 2 n + 4 }.
r
=
2
∑
n
r
(
r
+
2
)
1
=
12
5
−
2
n
+
2
1
−
2
n
+
4
1
.
[4]
(b)
Explain why
∑
r
=
2
∞
1
r
(
r
+
2
)
{\displaystyle \sum_{r=2}^\infty \frac{1}{r (r + 2)}}
r
=
2
∑
∞
r
(
r
+
2
)
1
is a convergent series, and state the value of the sum to infinity.
[2]
Answer
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