Binomial Sum!

Calculus Level 3

lim ⁡ n → ∞ ∑ k = 0 n 1 ( n k ) = ? \large \lim_{n\to\infty} \sum_{k=0}^n \dfrac1{\dbinom nk} = \, ?


The answer is 2.

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Abdelghani Ssoris by Abdelghani ßoris , 24, Morocco

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2 solutions

,

So:

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Chew-Seong Cheong
Sep 14, 2018

L = lim ⁡ n → ∞ ∑ k = 0 n 1 ( n k ) = lim ⁡ n → ∞ ( 1 + 1 n + 2 n ( n − 1 ) + ⋯ + 2 n ( n − 1 ) + 1 n + 1 ) = 2 \begin{aligned} L & = \lim_{n \to \infty} \sum_{k=0}^n \frac 1{\binom nk} \\ & = \lim_{n \to \infty} \left(1 + \frac 1n + \frac 2{n(n-1)} + \cdots + \frac 2{n(n-1)} + \frac 1n + 1\right) \\ & = \boxed 2 \end{aligned}

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