My Math Forum (-1)^n in single equality

 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 Display Modes Linear 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

 Contact - Home - Forums - Cryptocurrency Forum - Top