Expression of the day

I tried comb(2n,n) expressions. Many expressions on pi, square root of numbers and golden ratio were obtained. I used Wolfram Alpha to check(special thanks to Wolfram Alpha).

I tried Rational number series. Many expressions were obtained.

Today I decided to use both of them together. And what did I get ? Here we go . . . 

And here is the Wolfram Alpha screenshot.

I am deeply indebted to Wolfram Alpha. I mainly try mathematics only. Occasionally I also fire some Indian history queries too. It is amazing !!

Have a great day, dear blog readers !!

Back to back expressions

These are back to back expressions.

These are easy to verify. It is possibly 7th-8th class(India) material. Maybe some adults with not much of mathematics background may find them challenging. But I urge you to try them. You may have free time also due to COVID. Try these and verify.

But I feel the concept is BSc level. The concept of (mn+m-1)/(mn+m) - (mn-1)/(mn). Especially after so many (uncountable) results. Unfortunately they are not doctoral level. Otherwise I could have got a break. The gamma function result and the binomial results are advanced. But possibly only MSc level. But I am good in mathematics of electrical engineering. My basic qualification.

The expressions are 

Have a nice day, dear blog readers.

Twelve things that inspire my mathematics

I am writing an autobiography of sorts. 

There are twelve things that inspire my mathematics. These are

1. e^t. There are three different models of thinking I have on e^t.

2. 1/t!. I have done practical exercises to demonstrate the importance of 1/t!.

3. n/(n+1). I obtained an infinite series on it. This was my first major infinite expression.

4. (n+1)/n. Measuring behavior thought came up in 2010. (n+1)/n was an outcome of Measuring behaviour and also infinite series.

5. Comb(2n,n). Outcome of measuring behaviour. Two things. Fairness and unfairness.

6. Continued fractions. Vilvam (bael) tree was close to my thinking. I tried to represent Vilvam tree in continued fractions.

7.  (mn+m-1)/(mn+m)-(mn-1)/(mn). This expresses 3 consecutive numbers and 4 consecutive numbers concept. And more.

8. Geometric constructions. I was able to construct a pentagon and heptadecagon with a compass and straight edge.

9. Change produces area. My dafli playing teaches me this. Unfortunately my rhythm in calculus is not as strong as my rhythms in beats.

10. y to 1/y. I have a strong conviction here. God made integers and Pythagoras law becomes evident to me in this thread of thought.

11. Infinite series. Measuring behavior for fairness was made in 2010. And the concept of infinite series got laid.

12. #PythaShastri mantras. These are numeric utterances which makes harmony. Many of my #PythaShastri mantras have fetched results.

I thought of sharing my deep thoughts. Have a nice day dear readers !!

And now Gauss

Gauss made a brilliant equation. Integral of e to the power minus x squared for minus infinity to plus infinity equals square root of pi.

In place of x squared I put three consecutive numbers equation and four consecutive numbers equation. The result obtained is obedient. It is beautiful.

They are

I have used online Latex mathematical editor to type these.

Catalan's constant

I was working further on the complex function of mine.

I developed an expression. I tried it on Wolfram Alpha and the result show a term with Catalan's constant. It is indeed a privilege to get Catalan's constant in my expression.

Catalan's constant was first observed by Euler. However Charles Catalan did a lot of expressions and the constant is named after him. 

Below is the expression I made

Hooray !! A complex function !!

Made a complex function too. Here it is :

Have a nice day, dear blog readers !!

Hooray !! Integral and Sigma equivalence

 I have managed an integral and sigma equivalence. The equation is given below. The only other instance I know is from Parseval who did integral and sigma equivalence.

I am feeling ecstatic !!

Favorite four

 These are my favorite four expressions

You may try these for some random values and let me know if anything is wrong. I make errors inadvertently. So do let me know.

Have a nice day !!