May 13th, 2015, 09:01 AM  #1 
Newbie Joined: May 2015 From: India Posts: 2 Thanks: 0  Limits
[MATH] \lim_{x \to 0} \left [\frac{1}{1 \sin^2 x}+ \frac{1}{2 \sin^2 x} +....+ \frac{1}{n \sin^2 x}\right]^{\sin^2x} [\MATH] I took [MATH] \sin^2 x [\MATH] out of the brackets . Inside the brackets , I think I should use the formula [MATH] n(n1)/2 [\MATH] . Am I doing right ? If yes, then what should I do next ? Thanks ! 
May 13th, 2015, 09:16 AM  #2  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,031 Thanks: 2342 Math Focus: Mainly analysis and algebra 
To restate the problem: Quote:
However, writing $s_n = \sum_{k=1}^n \tfrac1k$ we have $$\left({s_n \over \sin^2 x}\right)^{\sin^2 x}$$ The numerator is heading to unity. So you need to investigate the limit of $x^{x}$ as $x \to 0$. This is most easily achieved by writing $x^{x} = \mathrm e^{x \log x}$.  

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