Recapitulating

Let me recapitulate my contributions in Rational number series :-

1. A base for binomial expressions.  

2. Phi, pi, e and gamma expressions. 

3. Scope for 3 consecutive numbers and 4 consecutive numbers pi expressions as a big theory.

4. Scope for consecutive numbers phi expressions as a big theory.

5. Scope for 3 consecutive numbers and 4 consecutive numbers in origami sheets for symmetry theory.

6. A few infinite series expressions. 

7. A few finite series expressions. 

I believe in openness. And sharing.  That is the crux of paper writing.  References. Anyone interested can use these ideas.

Let's hope for the best.  . . .

Mesmerizing

Alon Amit

Alon Amit is now in US. Born in Israel this man is wonderful and seem to have all answers in mathematics. He is a genius and I am following him in the site Quora. 

Marcello Barenghi 

Marcello Barenghi is an Italian and his paintings are outstanding. Top of the world talent. I am following him in YouTube. 

My good wishes to both of them.

The light bending

I can "see" the light bending.  Yes.

Remember my 1,x,x^2 and 1, 1/x,1/x^2 triangles.

Now 90 degrees is important.  It is of the field. Gravitational field.

Also 1,x,x^2 cannot be drawn on a 2D paper as right triangle.  However when 1,1/x,1/x^2 is considered you realize that 1/x^2 is smaller than 1/x making a triangle impossible.  Hence you realize it must be 1/x^2+1. So the Pythagoras theorem is seen

Now, also it a fact that Pythagoras theorem can be drawn on a 2D paper.

So what do we have? 1,x,x^2 which was not 2D becomes 2D with 1,1/x,1/x^2.

And . . . I have formulas of golden ratio and it's inverse.

All in all, these observations indicate that I can see light bending. 

With this, my knowledge is accepted across the globe.

If I travel to Japan,  Nigeria,  England, USA, Brazil or anywhere I shall be considered as a man of knowledge. 

A man of military or  sports ability or business ability or artistic ability is better. But something is better than nothing. 

Expression for gamma, Euler-Mascheroni constant

Euler-Mascheroni constant or gamma can also be obtained using the consecutive numbers theory.

Here is one

So, all in all using the consecutive numbers concept expressions for phi, pi, e and gamma have been obtained.

So far so good.

Have a very fine day, dear readers.

Expression for e, Euler's number

 Using the 3 consecutive number theory, one can get an expression for e the Euler's number.

Below an expression is given:-

Hope you like this expression. My best wishes to you !!

Gamma functions

 The three consecutive numbers and four consecutive numbers fetch beautiful gamma functions. 

Here are the Wolfram screenshots

You may recall that gamma functions are nothing but factorials.

Gamma function of n is (n-1)!

So essentially the first result is (2m+1)!. So the product of three consecutive numbers (in the above fashion ) taken in odds / or evens (depending on how one looks) give factorials.

It is interesting if we divide the results obtained above. What we get is an expression of pi. See the Wolfram result below:-

What I have done above is the division of the first result with the second result. Essentially the product of the three consecutive numbers and the product of four consecutive numbers have been divided. In the first two cases they were considered till m. In the third till infinity. We get an expression of pi.

This is the second expression for pi with 3 consecutive numbers and 4 consecutive numbers.

Hope you like these expressions. Have a fine day!!

Expression for 20th September 2021

 The below is the expression for 20th September 2021.

I hope you like it. Have a very fine day !!