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giti
giti
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I am trying to show that $$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\prod_{j=0}^{i}\frac{kn-j-k}{kn-j}=\frac{k^{k+1}-(k-1)(k-1)^{k}}{(k+1)k^{k}}$$$$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\prod_{j=0}^{i}\frac{kn-j-k}{kn-j}=\frac{k^{k+1}-(k-1)^{k+1}}{(k+1)k^{k}}$$ for all $k\in\mathbb{N}$, $k\geq 4$.

I could verify the statement with Mathematica, but I could not find a self-standing proof. The product is telescopic, but I could only derive lower bounds instead of the exact value of the limit.

I am trying to show that $$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\prod_{j=0}^{i}\frac{kn-j-k}{kn-j}=\frac{k^{k+1}-(k-1)(k-1)^{k}}{(k+1)k^{k}}$$ for all $k\in\mathbb{N}$, $k\geq 4$.

I could verify the statement with Mathematica, but I could not find a self-standing proof. The product is telescopic, but I could only derive lower bounds instead of the exact value of the limit.

I am trying to show that $$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\prod_{j=0}^{i}\frac{kn-j-k}{kn-j}=\frac{k^{k+1}-(k-1)^{k+1}}{(k+1)k^{k}}$$ for all $k\in\mathbb{N}$, $k\geq 4$.

I could verify the statement with Mathematica, but I could not find a self-standing proof. The product is telescopic, but I could only derive lower bounds instead of the exact value of the limit.

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giti
giti
  • 43
  • 4

Limit of the average of telescopic products

I am trying to show that $$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\prod_{j=0}^{i}\frac{kn-j-k}{kn-j}=\frac{k^{k+1}-(k-1)(k-1)^{k}}{(k+1)k^{k}}$$ for all $k\in\mathbb{N}$, $k\geq 4$.

I could verify the statement with Mathematica, but I could not find a self-standing proof. The product is telescopic, but I could only derive lower bounds instead of the exact value of the limit.