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 October 15th, 2019, 07:03 AM #1 Senior Member   Joined: Dec 2015 From: Earth Posts: 826 Thanks: 113 Math Focus: Elementary Math Limits with sequences Evaluate: a. $\displaystyle l=\lim_{N\rightarrow \infty } \underbrace{sinsin...sin}_{N}N$. b. $\displaystyle l=\lim_{n\rightarrow \infty } n!^{-2n}\prod_{i=1}^{n}i^i$. c. $\displaystyle l=\lim_{n\rightarrow \infty } \frac{\sum_{i=1}^{n^2 }i^{-1}}{\ln(n)}$. October 15th, 2019, 08:30 AM #2 Senior Member   Joined: Jun 2019 From: USA Posts: 380 Thanks: 205 a. is quite obviously zero. Thanks from idontknow October 15th, 2019, 10:17 AM   #3
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 Originally Posted by DarnItJimImAnEngineer a. is quite obviously zero.
I agree , $\displaystyle 0<\displaystyle \underbrace{sinsin...sin}_{N}N <1/N$ or $\displaystyle 0<l\leq \lim_{N\rightarrow \infty } 1/N=0$.
$0<l<0 , l=0.$ October 15th, 2019, 03:08 PM   #4
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 Originally Posted by idontknow I agree , $\displaystyle 0<\displaystyle \underbrace{sinsin...sin}_{N}N <1/N$ or $\displaystyle 0 Its true that the limit is zero but its not via comparison to$1/N$. The sequence$\sin \sin \sin \cdots \sin N$refers to function composition, not a product. So$-1 < \sin N < 1$is the last "obvious" step. For instance, in the second iteration you need to prove that$-\frac{1}{2} < \sin \sin N < \frac{1}{2}$and I don't see any reason that this is obviously true. October 16th, 2019, 12:26 AM #5 Senior Member Joined: Dec 2015 From: Earth Posts: 826 Thanks: 113 Math Focus: Elementary Math set N=2p and let$\displaystyle sin_{N}N =\underbrace{sinsin...sin}_{N}N$,$\displaystyle \displaystyle sin_N (N) < sin_{N-2}sin\frac{N}{2}
 October 16th, 2019, 04:07 AM #6 Senior Member   Joined: Dec 2015 From: Earth Posts: 826 Thanks: 113 Math Focus: Elementary Math b. $\displaystyle \: \frac{n^n }{n!^{2n} } \rightarrow 0 < \displaystyle n!^{-2n}\prod_{i=1}^{n}i^i < e^{-n^2 }\frac{n^{2n^2 }}{n!^{2n} }=e^{-n^2 } (\frac{n^{n}}{n! })^{2n}\rightarrow 0$. Tags limits, sequences 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 someone111888 Calculus 1 February 20th, 2018 11:20 PM Intube Calculus 2 October 1st, 2017 02:50 PM Intube Calculus 0 September 17th, 2017 11:54 AM BobaJ Real Analysis 1 March 28th, 2016 03:20 PM Luiz Real Analysis 3 May 11th, 2015 10:01 AM

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