Wednesday, January 27, 2016

sequences and series - limit limlimitsntoinftyleft(sumlimitsni=1frac1sqrti2sqrtnright)



Calculate below limit lim


Answer



As a consequence of Euler's Summation Formula, for s > 0, s \neq 1 we have \sum_{j =1}^n \frac{1}{j^s} = \frac{n^{1-s}}{1-s} + \zeta(s) + O(|n^{-s}|), where \zeta is the Riemann zeta function. In your situation, s=1/2, so \sum_{j =1}^n \frac{1}{\sqrt{j}} = 2\sqrt{n} + \zeta(1/2) + O(n^{-1/2}) , and we have the limit \lim_{n\to \infty} \left( \sum_{j =1}^n \frac{1}{\sqrt{j}} - 2\sqrt{n} \right) = \lim_{n\to \infty} \big( \zeta(1/2) + O(n^{-1/2}) \big) = \zeta(1/2).


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