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Limit of a product (TOMATO Subjective 157)

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

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

 

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May 10, 2014

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