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 May 13th, 2015, 08:01 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(n-1)/2 [\MATH] . Am I doing right ? If yes, then what should I do next ? Thanks ! May 13th, 2015, 08:16 AM   #2
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To restate the problem:
Quote:
 Originally Posted by EllaRosewood $\displaystyle \lim_{x \to 0} \left [\frac{1}{1 \sin^2 x}+ \frac{1}{2 \sin^2 x} +....+ \frac{1}{n \sin^2 x}\right]^{\sin^2x}$ I took $\displaystyle \sin^2 x$ out of the brackets . Inside the brackets , I think I should use the formula $\displaystyle n(n-1)/2$ . Am I doing right ? If yes, then what should I do next ? Thanks !
$\tfrac12 n(n-1)$ is the formula for the sum of the first $n$ natural numbers (including zero). This is a harmonic series.

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 $x^{-x} = \mathrm e^{-x \log x}$. Tags limit, limits Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post debad Calculus 4 July 16th, 2013 09:25 PM cheww Calculus 5 June 16th, 2013 11:03 PM justusphung Calculus 11 June 3rd, 2013 10:19 AM kadmany Calculus 9 March 18th, 2011 06:16 AM duckfan36 Calculus 2 October 3rd, 2010 01:49 PM

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