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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 : $\displaystyle (1)^n =\frac{n1}{2}+\frac{1}{\pi} \sum_{j=1}^{\infty} \frac{\sin(jn\pi)}{j}$. 
October 20th, 2019, 07:54 AM  #3  
Senior Member Joined: Jun 2019 From: USA Posts: 376 Thanks: 202  Quote:
$\displaystyle (1)^n$ 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 
$n\in \mathbb{N} \implies (1)^n = \cos(n\pi)$

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  $${\Huge{(1)^n = e^{i \pi n}}}$$ 
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. 

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