More punch to Rational Series

It is possible to make the Rational number series more accurate and more precise.

One needs to add or subtract geometric progression of some other number terms.

I tried it on my rational number series and have met with success.

I like representing numbers this way. As Rational number series. I find it more meaningful.

But mathematicians know about it. Gauss has done work. But he and others talk of it in some other way. Moreover, Gauss was too great and was able to "smoothen" expressions. I manage to get a rough expressions of Gauss. While he was "smooth" and "so graceful".

My mind gets clean bowled by the terrific expressions of Taylor, Gauss and Maclaurin in particular. I have very deep admiration for Abel. And am awed by Euler.

I have desire to make one beautiful expression. But that is not to be. But a few practical expressions I have attempted. These are okayish, I guess.