Elementary Math Fractions, Percentages, Word Problems, Equations, Inequations, Factorization, Expansion

 October 20th, 2019, 01:33 AM #1 Senior Member   Joined: Dec 2015 From: Earth Posts: 823 Thanks: 113 Math Focus: Elementary Math (-1)^n in single equality If anyone finds it useful here it is . $\displaystyle (-1)^n =4\lfloor n/2 \rfloor -2n+1 $$\displaystyle \; , n\in \mathbb{N} . October 20th, 2019, 05:51 AM #2 Senior Member Joined: Dec 2015 From: Earth Posts: 823 Thanks: 113 Math Focus: Elementary Math Here is the analytic expression : \displaystyle (-1)^n =\frac{n-1}{2}+\frac{1}{\pi} \sum_{j=1}^{\infty} \frac{\sin(jn\pi)}{j}. October 20th, 2019, 07:54 AM #3 Senior Member Joined: Jun 2019 From: USA Posts: 376 Thanks: 202 Quote:  Originally Posted by idontknow Here is the analytic expression : \displaystyle (-1)^n =\frac{n-1}{2}+\frac{1}{\pi} \sum_{j=1}^{\infty} \frac{\sin(jn\pi)}{j}. This gives \displaystyle \frac{n-1}{2} for any n\in \mathbb{N}. \displaystyle (-1)^n and \displaystyle e^{in\pi} are both analytic expressions. If you are trying to express it as a Fourier series, it only has one frequency: \pi. October 20th, 2019, 07:58 AM #4 Global Moderator Joined: Dec 2006 Posts: 21,105 Thanks: 2324 n\in \mathbb{N} \implies (-1)^n = \cos(n\pi) Thanks from Maschke, topsquark and idontknow October 20th, 2019, 04:38 PM #5 Senior Member Joined: Mar 2015 From: Universe 2.71828i3.14159 Posts: 169 Thanks: 64 Math Focus: Area of Circle$${\Huge{(-1)^n = e^{i \pi n}}}$$Thanks from idontknow October 21st, 2019, 01:13 AM #6 Senior Member Joined: Dec 2015 From: Earth Posts: 823 Thanks: 113 Math Focus: Elementary Math No need to check whether n is odd or even. Just plug any natural value of n and it gives the result . Same like expressing$\displaystyle (-1)^n \$ in terms of n avoiding the nature of number (odd,even)...etc . Edit: The expression may be useless but rare . Last edited by idontknow; October 21st, 2019 at 01:22 AM. Tags equality, single 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 Mr Davis 97 Algebra 1 August 14th, 2014 08:47 PM Messerschmitt Calculus 5 September 1st, 2012 12:11 PM Pavhard Number Theory 6 October 18th, 2010 11:24 PM hatcher777 Algebra 5 January 22nd, 2007 06:48 PM Pavhard Math Events 0 December 31st, 1969 04:00 PM

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