On Convergence

Convergence in mathematical infinite series was learnt by me on my own.

1+2+3+4+5+. . . . . . does not lead us anywhere. Does it?

1/1+1/2+1/3+1/4+1/5+. . . .  at least makes one consider numbers between 1 and 0. What a great piece of thought !! 

When I drew a pentagon, suddenly I thought of 1/4 in terms of 1/5,1/(5^2),1/(5^3) etc. I do not know where the thought came from. The rest is history, in my life. Expressions of infinite series began to take shape in me and I started my expressions of infinite. But I did do 22/7, e and Measuring Behaviour quite early. At least 2010.

1,-1,1,-1,1,-1 . . . . . . . .  is 1/2 in string theory. Because the result at any stage in this infinite series is either 0 or 1. And the average is 1/2. Using this result and one more does one come to realize that 1+2+3+4+5+. . . . . . =(-1/12).

How does one say a series converges? Using limits type of theory. Consider 55th item slightly increased from left hand side and 57th item slightly decreased from right hand side. Both should point at 56th item. 

If you want advanced theory in convergence, then there are theories on them.

I am more interested in 3 consecutive numbers, 4 consecutive numbers sort of things. 

Have a good day, dear blog readers !!

Cauchy Convergence test

Cauchy is known for brilliance. He got admission to Paris Polytechnic. He has a very important binomial theorem to his name. He comes second in the number of searches. Only after Euler.