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Non Integration Method

To find the summation formula for the $nth$ power $(n$ greater than $1$ ):

There will be $floor[\frac{n}{2}+2]$ terms to be written down.

Start by writing down: $x^{n+1}+x^n+x^{n-1}$

The remaining terms are derived by alternately subtracting then adding $x$ raised to a power $2$ less than the previous power.

To continue the algorithm we will show an example of writing out the formula for the summation of $x^6$

$x^7+x^6+x^5-x^3+x$

Our last step is to multiply each term by its appropriate coefficient, we will refer to these coefficients as the $z$ number, so the first coefficient is $z_1$ the second $z_2$ etc.

So the new formula will be: $z_{1}x^7+z_{2}x^6+z_{3}x^5-z_{4}x^3+z_{5}x$

The formulas for the first 5 $z$ numbers are:

$z_{1}(n)=\frac{1}{n+1}$

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