Monday, 30 June 2014

sequences and series - Finding limit of a product.



Prove:lim
I tried using Squeeze Theorem but can't go beyond $1

Answer



\frac{1}{n}\left[\prod_{i=1}^n(n+i)\right]^{1/n}=\left[\prod_{i=1}^n \frac{1}{n}(n+i)\right]^{1/n}=\left[\prod_{i=1}^n (1+\frac{i}{n})\right]^{1/n}

Taking log we get
\frac{1}{n}\sum_{i=1}^n\ln (1+\frac{i}{n}) \to \int_0^1 \ln(1+x)dx, n \to \infty
Integrating by parts gives \int_0^1 \ln(1+x)dx=\ln 4 -1.
Now the limit of the product is e^{\ln 4 - 1}.


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