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Question 0501a
Sigma Notation
Method of Differences I
Question
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Find, in terms of
n
,
{n,}
n
,
∑
r
=
1
n
(
1
r
!
−
1
(
r
+
1
)
!
)
.
\sum_{r=1}^n \Big( \frac{1}{r!} - \frac{1}{(r+1)!} \Big).
r
=
1
∑
n
(
r
!
1
−
(
r
+
1
)!
1
)
.
Attempt
The answer is of the form
1
(
n
−
1
)
!
−
a
{\frac{1}{(n-1)!} - a}
(
n
−
1
)!
1
−
a
1
n
!
−
a
{\frac{1}{n!} - a}
n
!
1
−
a
1
(
n
+
1
)
!
−
a
{\frac{1}{(n+1)!} - a}
(
n
+
1
)!
1
−
a
a
−
1
(
n
−
1
)
!
{a - \frac{1}{(n-1)!}}
a
−
(
n
−
1
)!
1
a
−
1
n
!
{a - \frac{1}{n!}}
a
−
n
!
1
a
−
1
(
n
+
1
)
!
{a - \frac{1}{(n+1)!}}
a
−
(
n
+
1
)!
1
where
a
=
{a=}
a
=
Answer
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