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2014 BMT Spring
8
BMT 2014 Spring - Analysis 8
BMT 2014 Spring - Analysis 8
Source:
January 6, 2022
functional equation
fe
algebra
Problem Statement
Suppose an integer-valued function
f
f
f
satisfies
∑
k
=
1
2
n
+
1
f
(
k
)
=
ln
∣
2
n
+
1
∣
−
4
ln
∣
2
n
−
1
∣
and
∑
k
=
0
2
n
f
(
k
)
=
4
e
n
−
e
n
−
1
\sum_{k=1}^{2n+1}f(k)=\ln|2n+1|-4\ln|2n-1|\enspace\text{and}\enspace\sum_{k=0}^{2n}f(k)=4e^n-e^{n-1}
k
=
1
∑
2
n
+
1
f
(
k
)
=
ln
∣2
n
+
1∣
−
4
ln
∣2
n
−
1∣
and
k
=
0
∑
2
n
f
(
k
)
=
4
e
n
−
e
n
−
1
for all non-negative integers
n
n
n
. Determine
∑
n
=
0
∞
f
(
n
)
2
n
\sum_{n=0}^\infty\frac{f(n)}{2^n}
∑
n
=
0
∞
2
n
f
(
n
)
.
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