Note: I wrote the above because I ran into the rather important series expansion at another point. I tried to look into my notes & I had barely noted McLaurin's series in them (no subheading & no explanations), binomial series expansion was also in another form & unsatisfactory. I looked at the textbooks I had & they did not explain properly the series expansion. Then I looked online & people were dismissing similar queries with "its a binomial series expansion ... " etc. It was not immediately obvious to me how the expansion was taking place & I had a feeling of unease about it. So I sat down to expand it myself & wrote it down for others to view.
It is far better to write or do or think something worthwhile that improves/ perfects the quality of life of all than to write some arcane mathematical jargon that is not accessible to all but the "educated" (& hopefully gentrified, entirely good, virtuous & righteous, though there seem to be no guarantees, unfortunately).
