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October 20th, 2019, 01:33 AM   #1
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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} .$
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October 20th, 2019, 05:51 AM   #2
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Here is the analytic expression :
$\displaystyle (-1)^n =\frac{n-1}{2}+\frac{1}{\pi} \sum_{j=1}^{\infty} \frac{\sin(jn\pi)}{j}$.
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October 20th, 2019, 07:54 AM   #3
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Originally Posted by idontknow View Post
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: $\pi$.
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October 20th, 2019, 07:58 AM   #4
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$n\in \mathbb{N} \implies (-1)^n = \cos(n\pi)$
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October 20th, 2019, 04:38 PM   #5
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$${\Huge{(-1)^n = e^{i \pi n}}}$$
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October 21st, 2019, 01:13 AM   #6
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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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