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Test of Mathematics Solution Subjective 157 -Limit of a product

Test of Mathematics at the 10+2 Level

This is a Test of Mathematics Solution Subjective 157 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.


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Problem

Evaluate $ \mathbf {\displaystyle \lim_{n \to \infty } \{ (1 + \frac{1}{2n}) (1 + \frac{3}{2n} )(1+ \frac{5}{2n})  ...  (1+ \frac{2n-1}{2n})\}^{\frac{1}{2n}} }$


Discussion

Let $ \mathbf { y = \{ (1 + \frac{1}{2n}) (1 + \frac{3}{2n} )(1+ \frac{5}{2n})  ...  (1+ \frac{2n-1}{2n})\}^{\frac{1}{2n}} }$

Then $ \mathbf { \log (y) = \frac{1}{2n}\{{ \log (1 + \frac{1}{2n}) + \log (1 + \frac{3}{2n} )+ \log (1+ \frac{5}{2n}) + ... + log (1+ \frac{2n-1}{2n})} }\}$

This implies $ \log (y) = \frac{1}{2n} { \log (1 + \frac{1}{2n}) + \cdots + \log (1 + \frac{2n}{2n}) } $ - $ \frac{1}{2n} { \log (1 + \frac{2}{2n}) + \log (1 + \frac{4}{2n} )+ \log (1+ \frac{6}{2n}) + ... + \log (1+ \frac{2n}{2n}) } $

$ \displaystyle \lim_{n \to \infty} \log (y) = \int_0^1 \log (1+x) dx - \frac{1}{2} \int_0^{1} \log(1+x) dx = \frac{1}{2} \int_{0}^1 \log (1+x) dx = \log 2 - \frac {1}{2} $

Therefore answer will be \(e^{\log 2 - \frac {1}{2}}\)

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