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. 

Pebble smartwatch paired

Pebble is closed. I managed to get watchface earlier with difficulty when I accidentally reset my earlier Samsung J2 phone.

I got for myself Samsung A31. Finally after two days I got my old Pebble smartwatch to pair with modern Samsung A31.

More stuff in mathematics

Gauss had this brilliant idea of complex functions. The three thing.  The real axis. The imaginary axis. And you. You use the function.  

There is nothing as brilliant as the prime number stuff from Euler, my master.

I followed Madhava of Sangamagrama because of nuptial liberty. I would not have tried pi otherwise.  And I got an expression in pi of root 3 fashion when I followed Madhava. 

Am I clear in pi? Fairness in measuring  behavior was there. Unfairness no. I realized unfairness and came up with comb(2n,n). It was impossible to be fair and unfair both. If this is impossible I wonder how 1 by pi. And how comb(2n,n) cubed. When comb(2n,n) would  have been sufficient. Truly Ramanujan was a genius. He showed  the neutrality of gamma function  by using it in numerator and denominator. Truly outstanding!!

(Added on 11th January 2021)

Stuff in mathematics

The very basic thing is the functions when approaching mathematics. When you say function then it has some value say for 1,2,3 and also perhaps -1,-2,-3. The concept of limits and continuity arises. Newton struck his unsurpassable genius here. He made calculus out of limits. Rest is history.

There is one point I would like you to think about. You say this is an apple. This is an orange. Don't you have doubts? Shouldn't you have doubts to be so certain? Are you perfect? Should you not correct yourself and express yourself better? Welcome, Gauss. There are two views. And you become the third. 

Newton seems correct because of presence of field. Gravity or others. Which seem to make numbers. Hence the function. And so the calculus.

So are there numbers or not? What about primes? I am confused here. The prime numbers thing. Euler made such powerful equations on primes. And Euler totient function too.

This is what I understand. This is what I feel.

Have a great day, dear readers !!

Made it once again

Yes. I have done it once again. I have taken the help of Wolfram Alpha for this. I have not checked exhaustively. But am reasonably sure of them. Very very special thanks to Wolfram Alpha. I could not have expressed myself mathematically without Wolfram Alpha.

Here we go . . . 

The first expression is a finite series expression. The second expression has binomial expression.

Hope you like them.

Once again, I wish you a very nice day, dear blog readers. May you be happy with all your family members.

The Other Way

 Consider the following.

I have been trying expressions from (mn+m-1)/(mn+m)-(mn-1)/(mn) till now. There is another way to say this. And that is (mn)/(mn+1)-(mn-m)/(mn-m+1).

In coming days, I shall try to explore expressions from (mn)/(mn+1)-(mn-m)/(mn-m+1). Only finite and binomial statements. I shall be blogging if I am able to make finite series and binomial statements.

Till then Goodbye. And Jai Shri Krishna.

I can be here

I can be here. Yes, serious !!

It is just some more that is required. 

An expression like the one below can be made with the help of Wolfram Alpha

But I want to do service to my present organization. I have pride working in SAIL because of the high quality of stainless steel that it produces. But there are outstanding problems too. Severe competition from Tatas, Jindals and others. And also customers who may opt for imported steel too if quality is good.

SAIL brand has many faithful followers. This fact will surely help us clear a lot of our loans. I believe in it. And 'SAIL will make it happen'.

Khuda hafiz.