Quiz 8 — March 16, 2011 – Section 4 – 9:059:55 recitation.
.
Consider the series
∞
∑
n
=2
2
n
2

1
.
(a) Let
M
≥
2 be some fxed integer. Find a closed ±ormula ±or the partial sum
s
M
=
M
∑
n
=2
2
n
2

1
. (Comment. It is possible to use the technique o± partial ±ractions
to express this series as a telescoping series.)
(b) What is the sum o± the series?
Answer.
Write
2
n
2

1
=
A
n

1
+
B
n
+1
. Multiply by
n
2

1 to see that
2 =
A
(
n
+ 1) +
B
(
n

1)
.
Plug in
n
= 1 to see that 1 =
A
. Plug in
n
=

1 to see that
B
=

1. This works
becasue
1
n

1
+

1
n
+ 1
=
n
+ 1

(
n

1)
n
2

1
=
2
n
2

1
.
(a) We see that
s
M
=
M
s
n
=2
2
n
2

1
=
M
s
n
=2
b
1
n

1

1
n
+ 1
B
=
±
1
2

1

1
2+1
²
+
±
1
3

1

1
3+1
²
+
±
1
4

1

1
4+1
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 Fall '11
 KUSTIN
 Calculus, Fractions, Summation, Partial sum, closed formula, Answer. Write

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