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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
=
4
n
1
(
r
+
1
)
(
r
+
3
)
=
11
60
−
1
2
n
+
4
−
1
2
n
+
6
.
\sum_{r=4}^n \frac{1}{(r + 1) (r + 3)} = \frac{11}{60} - \frac{ 1 }{ 2 n + 4 } - \frac{ 1 }{ 2 n + 6 }.
r
=
4
∑
n
(
r
+
1
)
(
r
+
3
)
1
=
60
11
−
2
n
+
4
1
−
2
n
+
6
1
.
[4]
(b)
Explain why
∑
r
=
4
∞
1
(
r
+
1
)
(
r
+
3
)
{\displaystyle \sum_{r=4}^\infty \frac{1}{(r + 1) (r + 3)}}
r
=
4
∑
∞
(
r
+
1
)
(
r
+
3
)
1
is a convergent series, and state the value of the sum to infinity.
[2]
Answer
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