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10
2010 Calculus #10: Summation with Constants
2010 Calculus #10: Summation with Constants
Source:
July 15, 2012
calculus
Problem Statement
Let
f
(
n
)
=
∑
k
=
1
n
1
k
f(n)=\displaystyle\sum_{k=1}^n \dfrac{1}{k}
f
(
n
)
=
k
=
1
∑
n
k
1
. Then there exists constants
γ
\gamma
γ
,
c
c
c
, and
d
d
d
such that
f
(
n
)
=
ln
(
x
)
+
γ
+
c
n
+
d
n
2
+
O
(
1
n
3
)
,
f(n)=\ln(x)+\gamma+\dfrac{c}{n}+\dfrac{d}{n^2}+O\left(\dfrac{1}{n^3}\right),
f
(
n
)
=
ln
(
x
)
+
γ
+
n
c
+
n
2
d
+
O
(
n
3
1
)
,
where the
O
(
1
n
3
)
O\left(\dfrac{1}{n^3}\right)
O
(
n
3
1
)
means terms of order
1
n
3
\dfrac{1}{n^3}
n
3
1
or lower. Compute the ordered pair
(
c
,
d
)
(c,d)
(
c
,
d
)
.
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