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 n^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}

z_{2}(n)=\frac{1}{2}

z_{3}(n)=(n-0)(\frac{1}{12})

z_{4}(n)=(n-2)(n-1)(n-0)(\frac{1}{720})

z_{5}(n)=(n-4)(n-3)(n-2)(n-1)(n-0)(\frac{1}{30240})

Multiplying everything out gives us the correct formula of: \frac{x^7}{7}+\frac{x^6}{2}+\frac{x^5}{2}- \frac{x^3}{6}+\frac{x}{42}

Leave a comment