Consider when n=3, and i=3, yields:

Let

andthen;

let

Then

this equation is solvable.

Now let n=2

Let

andthen

There is no whole number solution according to Euler.

]]>Let the p-th power of an alternating arithmetic series as follows

The General Equations are given as follows:

For odd power:

For even power:

Where

Now, let consider n=3,

When n=3, the sums of power for p=3 reduces into:

Let

Then

Or

Now consider this equation

Where

and

Assuming w is an even, thus

and

Therefore

Solving the equation yields

Let s=1, then

w=12

Solving the equation yields,

, andand z=2a=6

Therefore, there is a solution for this equation, which is given as follows

Now consider when n=2 and using the same procedure.

When n=2, the sums of power for p=3 reduces into:

Let

Then

Assuming w is an even, thus

and

Solving the equation yields:

since w=2a=2(s/2)=s,

This is a trivial solution,

or

Consider

and

Solving the equations yields:

Imaginary Solution.]]>Let the expansion of (x+y) as follows:

Now let the T-th term of arithmetic progression as (a+bi).

Thus,

Summing the terms above yields:

Example:

=>

=>

As the p is getting larger, the calculation would be becoming tedious.

]]>You had an extra \left in there.

Btw, I would imagine it to be O_{m,k}, because the value of j in the expression isn't really constant...

]]>For odd power:

For even power:

Basically, there are many Bernoulli's formulations and the finding of new Bernoulli's formulation not that significant. In my paper, the development of new bernoulli's formulation is only a small portion and without this formulation I can get others formulation to get the numbers. Since, Sums of power got bernoulli's numbers in it, I managed to manipulate it to get new forms of bernoulli's formulation but the main purpose is to develop sums of power for arithmetic progression. I do believe finding new sums of power for arithmetic progression is a big thing before people get to know it. It can be used for numerical analysis, Riemman's zeta function, Fermat's Last Theorem, generating function for finding prime numbers etc. I had demonstrated few examples of the use of this formulation.

Fulhaber is known as one of the greatest mathematicians because he developed sums of power for integers. This encourage me to work on bringing this formulation to the world as it is the umbrella for all sums of power because it can do integers, non-integers, integer power, complex power and many more.

Here the bernoulli's formulation:

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