A general expression

Here is what seem to be happening in the expressions of the earlier blog post. m is assumed positive integer.

Have a fine day !!

More on Euler number

Remember my 1/e expressions and their extensions. 

If you don't then here is a screenshot.

I tried the reciprocal of above and one gets expressions of e. This is great.

Here are the screenshots.

Enjoy your day!!

Let us go back a little

 Do you remember my approximate expression for five consecutive numbers? If not, here is what I had to say

I rated it as very highly because when there is squares in five consecutive numbers, there is 2 in the numerator of RHS; and when there is square root 1/2 in the denominator.

However the common terms could be expressed like as shown below

I wanted to find out what is the expansion till infinity of the just above expression. I used Wolfram and the result is

So, my five consecutive numbers expressions of squares and square roots has mathematics to it.

Well that's all folks for now. Have a great day !!

A new expression for Golden Ratio

 Below is the Wolfram Alpha Screenshot

Hope you like this one. Putting y as -0.66666666666 or -2/3 gives n as phi.

Have a nice day !!

Made a great equation

 I have made a great equation using the help of Wolfram Alpha.

Good day wishes !!

Hip hip hooray !!

 Guys, I got a simpler expression for e with the Rational number series or consecutive numbers. Here it is

I was thinking that e is tougher with Rational number series and I had this expression earlier.

As you can see I was using a complex expression earlier. Now it is simpler.

Below is the Wolfram screenshot

Have a nice day ahead !!

(Added on 03/12/2021)

I am using the following 1/2 values to see whether the topmost expression can be studied with Rational Number Series or consecutive number theory or not. No doubt the topmost expression has a great value and I intend to develop further based on Rational number Series or consecutive number theory.

How is my idea ? Have a great day !!

Added 04/12/2021

Consider these below. I hope you see a pattern.

I am able to see the pattern. If you do too, then comment in this blog post below.

Together we shall make the world better by 5%. Yes. That is the level of beauty in this.

Have a wonderful day !!

Beauty of Euler number

Euler number e is so beautiful. It changed our world by 25%.

And given below are two expressions which say by themselves why e(Euler number) is so beautiful.

When the 2^m term is in denominator root of e is obtained. When 2^m is in numerator e squared is obtained. 

Here are the screenshots.

It is a great honor that I had approximate expressions with 2 in numerator for squared of five consecutive numbers and with 2 in the denominator for square root of five consecutive numbers.

This is what I am talking about.

So Euler number and my expressions have similar property.

Euler you were the greatest !!

Have a nice day, dear blog readers !!