# Tagged Questions

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### Let $G_n$ denote the geometric mean of binomial co-efficients… [duplicate]

Denote by $G_n$ the geometric mean of the binomial co-efficients $${n\choose 0},{n\choose 1},\ldots ,{n\choose n}$$ Prove that $$\lim_{n\to \infty}\sqrt[n]{G_n}=\sqrt e$$ My work: We have, ...
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### Limit of $\sum_{i=1}^n \left(\frac{{n \choose i}}{2^{in}}\sum_{j=0}^i {i \choose j}^{n+1}\right)$

I'm trying to calculate the limit for the sum of binomial coefficients: $$S_{n}=\sum_{i=1}^n \left(\frac{{n \choose i}}{2^{in}}\sum_{j=0}^i {i \choose j}^{n+1} \right).$$ Numerically I can see this ...
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### Simple Random Walk

How to find: $\lim\limits_{N \to \infty}\sum\limits_{m=0}^Nu_m$ where $u_m$=${2m \choose m}p^mq^m$ I know there are two cases to consider depending if $p$ and $q$ are equal or not. I should probably ...
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### Limit of a Sum Involving Binomial Coefficients

I would like to prove that $$\dfrac{\sqrt{n}}{{{2n \choose n}^2}} \cdot \sum_{j=0}^n {n \choose j}^4$$ converges to $\sqrt{\dfrac{2}{\pi}}$ as $n \to \infty$. Evaluating the sum in Matematica for ...
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### Find $\lim_{n\to\infty}$ of this quotient.

Find, with proof, the value of this limit $$\lim_{n\to\infty}\frac{\sum^n_{r=0}\binom{2n}{2r}\cdot2^r}{\sum^{n-1}_{r=0}\binom{2n}{2r+1}\cdot2^r}$$ I have tried using binomial identities but two ...
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### The $n^{th}$ root of the geometric mean of binomial coefficients.

$\{{C_k^n}\}_{k=0}^n$ are binomial coefficients. $G_n$ is their geometrical mean. Prove $$\lim\limits_{n\to\infty}{G_n}^{1/n}=\sqrt{e}$$
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### Limit of alternating sum with binomial coefficient

I need to find a limit, or approximation for $\sum\limits_{k=1}^{n} (-1)^k {n \choose k} \log(a+bk)$ for, say, an $a,b\in (0,10)$. It is not so important what values $a$ and $b$ have. It would be ...
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My question is: What is the result of this limit: $\displaystyle \lim_{n \to +\infty} \frac{{n \choose n/2}}{2^n}=$ ?
### Proofs of $\lim\limits_{n \to \infty} \left(H_n - 2^{-n} \sum\limits_{k=1}^n \binom{n}{k} H_k\right) = \log 2$
Let $H_n$ denote the $n$th harmonic number; i.e., $H_n = \sum\limits_{i=1}^n \frac{1}{i}$. I've got a couple of proofs of the following limiting expression, which I don't think is that well-known: ...
Do you think the following limits are correct? $\displaystyle\lim_{d\to\infty}\frac{\sum\limits_{k=1}^{d} {\varphi(N) \choose k} {d-1 \choose k-1}}{\varphi(N)^d}=0$ ...