1+2+3+4+ . . . do not lead us anywhere. Does it? The numbers keep increasing. The sum keeps increasing. It doesn't converge.
1/1+1/2+1/3+1/4 . . . also don't converge.
1/(1^2)+1/(2^2)+1/(3^2)+1/(4^2) . . . converges. It is the Basel problem. Euler, my master, proved it is equal to pi squared by 6.
1/(1^3)+1/(2^3)+1/(3^3)+1/(4^3) . . . converges. It is Apery constant.
These are of mathematics. I am not an expert but I made Measuring Behavior. I believe it is close to Wallis pi.
But I had a difficult time. Learning infinite series. I made #Pythashastri mantras and body expressions. I wrote them in a simplified way in Wolfram alpha. And I learnt my mathematics that way.
(Indebted to Wolfram alpha. My koti koti namaskarams)
I believe I have convergence test. A thumb rule sort.
I have conducted three experiments. Lifting dumbbell with left hand, lifting okras with fingers and observing the thickness of an object at relative distance. They gave me 1, gamma function and infinite arithmetic geometric progression. I have one more idea of reflex and lumens. But this cannot be done at home and by myself. These experiments give convergent values.
Okay. So far, so good.
Today is the world of software. Indians are deeply involved in software. We must develop models on pattern matching and get convergent value. This will be very useful and many of my convergent expressions of mine on pi, e, golden ratio and all will be benefitted.
I wish all my blog readers a pleasant day ahead !!
2 comments:
I have three things to say on my pi by 3 root 3 result.
One
Roger Penrose tiles based on phi result in approximate circular design
Two
I have golden ratio based pi formulas. See this
Three
Rational number series or consecutive numbers closely follow the Taylor Series model of
1
10
100
1000
10000
. . . and my pi by 3 root 3 are of rational number series or consecutive numbers.
It took Leonhard Euler 3 to 4 years to prove Basel problem. So proving the pi by 3 root 3 is likely to be difficult.
Therefore, in view of all the above pattern matching equivalence of convergence tested series must be resorted to
to get the convergent value
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