Taylor series and Rational number series

There is a very compelling and interesting similarity in Taylor series and Rational number series.

It is indeed a great honor that Rational number series has a similarity with Taylor series.

Please refer to below material :-

The topmost statement is the Taylor series.

The second statement is Rational number series expression.

The third is deduced from the second expression.

The fourth has been modified from the third expression based on a mathematical expression I made and posted earlier.