I suggest further use of reciprocal and test for triangle property. This step helps in understanding why sigma till infinity of 1 by n diverges. And why sigma till infinity of 1 by n factorial converges.
Basically I am influenced by triangle and reciprocal triangles. So I am suggesting these methods. You may recall that I experience convergence naturally and have done experiments too.
I further applied the convergence test of mine to 1/comb(2n,n) series and the pi formula of Madhava of Sangamagrama. The formula works satisfactorily Madhava of Sangamagrama formula was tricky as it had alternate pluses and minuses. The logic worked still.
I had to increase both the second term and the fourth term. Because it was moving from negative to positive in both when considering the third term (Madhava series). Try this formula for yourself. Who knows we may have a winner technique in this method !!
However, the formula did not work in 1/3-1/3squared+1/3cubed-1/3tothepowerfour. So, there are errors in the formula. Or perhaps how to treat alternate pluses and minuses.
Got it !! Geometric progression. No problem. For till infinity. 'Cos Gauss already had given a formula !! This finishes the topic. So what do we have? If geometric progression till infinity, then use Gauss formula for result and convergence. If otherwise, you may try my formula. Seems okay. Discussion over from my side.
What is the value of convergence of an infinite series? If you want my explanation then here are the links to my olden day ideas. Look at Euler Identity 1 and Euler Identity 2 and Basel Problem.
I shall probably be thinking of using these statements to know the value of convergence:- 1. Use finite series values. 2. Consider using n as pi and 2n as pi squared. 3. Consider a/b as rational where a and b are limbs in a right triangle. 4. Consider using area of a perfect right triangle as 0. 5. Use my expressions of 1 for 1 in series. 6. Deliberately add expressions of 0.
I am learning convergence from Khan academy. Meanwhile I checked my theory for two more series. 1/(n^n) and 1/(7n). Showed convergence for 1/(n^n). Which is true. Showed divergence for 1/(7n). Which is true.
13 comments:
I suggest further use of reciprocal and test for triangle property.
This step helps in understanding why sigma till infinity of 1 by n diverges. And why sigma till infinity of 1 by n factorial converges.
Basically I am influenced by triangle and reciprocal triangles. So I am suggesting these methods.
You may recall that I experience convergence naturally and have done experiments too.
I have written whatever I thought on convergence. This is over from my side.
Preliminary observations point that my theory works.
I work for my own.
I don't expect help.
Kindly point out if there are any errors!!
The formula works and tests the convergence of inverse fibonacci numbers and inverse triangular numbers too.
I believe this theory of mine works !!
I further applied the convergence test of mine to 1/comb(2n,n) series and the pi formula of Madhava of Sangamagrama.
The formula works satisfactorily
Madhava of Sangamagrama formula was tricky as it had alternate pluses and minuses. The logic worked still.
I had to increase both the second term and the fourth term. Because it was moving from negative to positive in both when considering the third term (Madhava series).
Try this formula for yourself. Who knows we may have a winner technique in this method !!
However, the formula did not work in
1/3-1/3squared+1/3cubed-1/3tothepowerfour.
So, there are errors in the formula. Or perhaps how to treat alternate pluses and minuses.
Got it !!
Geometric progression. No problem. For till infinity.
'Cos Gauss already had given a formula !!
This finishes the topic.
So what do we have?
If geometric progression till infinity, then use Gauss formula for result and convergence.
If otherwise, you may try my formula. Seems okay.
Discussion over from my side.
What is the value of convergence of an infinite series? If you want my explanation then here are the links to my olden day ideas.
Look at Euler Identity 1 and Euler Identity 2 and Basel Problem.
I shall probably be thinking of using these statements to know the value of convergence:-
1. Use finite series values.
2. Consider using n as pi and 2n as pi squared.
3. Consider a/b as rational where a and b are limbs in a right triangle.
4. Consider using area of a perfect right triangle as 0.
5. Use my expressions of 1 for 1 in series.
6. Deliberately add expressions of 0.
Kindly take a look at Tutorials on convergence at Khan academy for proper and methodical learning on convergence and what is the convergent value.
I am learning convergence from Khan academy.
Meanwhile I checked my theory for two more series.
1/(n^n) and 1/(7n).
Showed convergence for 1/(n^n). Which is true.
Showed divergence for 1/(7n). Which is true.
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