We will derive summation of series with increasing difficulty.

Lets derive the simplest series summation formulas,

To derive a formula to this series consider adding the same series with all numbers reversed, that is and so on till i.e, check the following equations,

Simplifying, we get

Well that’s a simple series how about sum of squares! before that we will understand simple properties of symbol.

If we observe the formula , the summation of is having terms of and alone, so if we do we will end up in equation with terms which can be kept on one side and rest of the terms will be either algebraic function of or only.

Lets define a series called S so that,

Also the series S expands as follows,

Now using above two equations of summation S, we have

By using this we can now solve sum of series like mentioned in the above example. Further we can continue with the same technique to get sum of cubic numbers,

Also the series S expands as follows,

Now using above two equations of summation S, we have

Hope you will go further to derive higher order formula and who knows a general solution!

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