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Putnam
1958 November Putnam
B1
Putnam 1958 November B1
Putnam 1958 November B1
Source: Putnam 1958 November
July 19, 2022
Putnam
binomial coefficients
limit
Problem Statement
Given
b
n
=
∑
k
=
0
n
(
n
k
)
−
1
,
n
≥
1
,
b_n = \sum_{k=0}^{n} \binom{n}{k}^{-1}, \;\; n\geq 1,
b
n
=
k
=
0
∑
n
(
k
n
)
−
1
,
n
≥
1
,
prove that
b
n
=
n
+
1
2
n
b
n
−
1
+
1
,
n
≥
2.
b_n = \frac{n+1}{2n} b_{n-1} +1, \;\; n \geq 2.
b
n
=
2
n
n
+
1
b
n
−
1
+
1
,
n
≥
2.
Hence, as a corollary, show
lim
n
→
∞
b
n
=
2.
\lim_{n \to \infty} b_n =2.
n
→
∞
lim
b
n
=
2.
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