A bit of Math

Life goes on, bad as ever. Last week i was wondering how things could get any worse. Lo! Down i went with fever and cough and cold and all that nonsense. Apparently there is a family of Streptococci residing in my throat who behave as though they own the place- or so my Doctor says, who also predicted a piles epidemic in India some twenty years ahead when all our IT fellas will come of age, having spent a lifetime squatting in front of monitors.

The fever subsided yesterday and i didn't know what to do so i did some math. It's not usual behaviour from me, and i suspect the fever-induced delirium had something to do with it. I was very worried about the number pi, i wanted to derive it from first-principles. This led to the below stated four sequence/series expressions for pi. I think the correct expression would be to say that i derived these, but i prefer the expression discovered.

1.


2.
i.e, to put it more simply

3.

and 4.



or, in simpler terms

(NB: The trigonometric functions sine,cos and tan given above have their arguments in degrees not radians)

So, are these new? I guess not. If I were born in, say, 17th century France, then probably I'd have stood a good chance to copyright the stuff. But had I been born in 17th Century France, I might never have bothered to study Math because i'd be bent in the pursuit of arts.Hmm... I love France, especially in the 17th century.

Alternative titles that I considered for this piece:

Genius Reveals Himself
Ramanujan II
Pi and Piles
The 'Pi'nk Panther
'Pi'yye thinna panayum thinnaam

PS: I Love You ;-)