Reciprocity is Golden Ratio supported

Here's how.

Hope you have a nice day!!

Infinite series for Golden Ratio

Here is another infinite series for Golden Ratio. With this I have made 3 infinite series expressions for golden ratio or phi.

Have a nice day, dear readers !!

The first infinite series

The second infinite series

And this equation is the third.

Thumb

Here are some expressions which I claim explains the metaphor of a thumb.

Have a very nice day day, dear readers!!

Thumb metaphor and pi and golden ratio (phi)

Beauty of Nature -II ; Reciprocity expressed

The beauty of nature which was posted in my earlier blog post is demonstrated here in terms of pi.

Just imagine, how beautiful nature is !! There is a precision in the plan of nature and the Lord above has accurately made everything to a plan.

If you observe carefully, both the expressions deliver the same modulus values. However one of them gives a negative result while the other positive result. And there is a change in the values of LHS and the change has been done as per the previous blog post.

Have a great day, dear readers !! I am sure you find these expressions exciting as I do. Further I am sure you appreciate the working of nature plan.

Beauty of Nature

 

Have a nice day, dear blog readers !!

Convergence test

As I experienced convergence I had to discover why some expressions converge and why some doesn't.  

I made a formula or a method for convergence. 

I am an inventor and made two unbeatable music methods. One is in rhythms.  And the other frequencies.  Unbeatable because the rhythm technique using hands is actually similar to walking technique.  And walking is so natural.  And in the other the hand is trained to move according to frequencies.

But Indians are different.  Thou shalt not invent. Thou shalt only follow USA and UK. Thou shalt not defeat the Chinese. Thou hath a place in heaven if thou practice JEE problems. Thou shalt not question seniors. Indians think like this.

Anyways. 

I am attaching a paper on convergence test. Though this paper is a static one, the vision of mine is dynamic and fast. Basically what I mean is if you increase the 4th term(or for that matter 16th term) and decrease the 6th term; and if a simple triangle 🔺️ property exists for the three numbers 4,5 and 6 or their reciprocals, then the expression convergence. 

Now Indians will say. Thou hath made a decent expression perhaps. But thou shalt give it free. Thou shalt not get training or exposure.

And Europeans just love ❤ these Indian thoughts!!

A general pi pattern

I have attempted a general pi pattern.

Hope you like it !! Have a nice day!!

Replace 2^n with 2^k.

The RHS will change. However the pattern is not yet established. I don't have a pattern.

Added on March 5, 2022

This expression below gives a relationship to what I have been trying. The result on RHS are in the form of gamma functions and root pi.