I had made five Rational Series expansions. These were:-
1. n/(n+1) - (n-1)/n = 1/n = 1/(n+1)+1/(n+1)²+1/(n+1)³+1/(n+1)⁴+ . . . . . .
2. The sequence n/(n+1) - (n-2)/(n-1) and the connection this series had with triangle series. The growth of this series was like 1/(4 * triangle series term).
3. The numerator i.e (n - √(n+1)*√(n+1)) of the expression √n/√(n+1) -√(n+1)/√n was approximately 1/(2*n)
4. n²/(n+1)² - (n-1)²/n² is approximately 2/(n+1)²
5. n²/(n+1)² - (n-2)²/(n-1)² is approximately prime / (n*4)². This was the prime number generator and more than 30% of the results were prime.
Today, I made the sixth expression. And that is
6. √ n/√ (n+1) - √ (n-2)/√ (n-1) is approximately 1/n².
So six expressions have been made. It may be confusing for a normal reader of my blogs. But these may be important. I have called then # expressions.
I am able to better my rational series expressions to be more accurate. However Euler totient functions exists. Euler totient functions are better expressions than rational series expressions. I have no doubts about it.
More accurate rational expressions are:-
1. The numerator i.e (n - √(n+1)*√(n+1)) of the expression √n/√(n+1) -√(n-1)/√n is more accurately 1/(2*n) + 1/(8*(n+1)³)
2. n²/(n+1)² - (n-1)²/n² is more accurately 2/(n+1)² + 1/(n+1)⁴
3. √ n/√ (n+1) - √ (n-2)/√ (n-1) is more accurately 1/n² + 1/(2.n³).
This finishes my mathematical endeavors.
1. n/(n+1) - (n-1)/n = 1/n = 1/(n+1)+1/(n+1)²+1/(n+1)³+1/(n+1)⁴+ . . . . . .
2. The sequence n/(n+1) - (n-2)/(n-1) and the connection this series had with triangle series. The growth of this series was like 1/(4 * triangle series term).
3. The numerator i.e (n - √(n+1)*√(n+1)) of the expression √n/√(n+1) -√(n+1)/√n was approximately 1/(2*n)
4. n²/(n+1)² - (n-1)²/n² is approximately 2/(n+1)²
5. n²/(n+1)² - (n-2)²/(n-1)² is approximately prime / (n*4)². This was the prime number generator and more than 30% of the results were prime.
Today, I made the sixth expression. And that is
6. √ n/√ (n+1) - √ (n-2)/√ (n-1) is approximately 1/n².
So six expressions have been made. It may be confusing for a normal reader of my blogs. But these may be important. I have called then # expressions.
I am able to better my rational series expressions to be more accurate. However Euler totient functions exists. Euler totient functions are better expressions than rational series expressions. I have no doubts about it.
More accurate rational expressions are:-
1. The numerator i.e (n - √(n+1)*√(n+1)) of the expression √n/√(n+1) -√(n-1)/√n is more accurately 1/(2*n) + 1/(8*(n+1)³)
2. n²/(n+1)² - (n-1)²/n² is more accurately 2/(n+1)² + 1/(n+1)⁴
3. √ n/√ (n+1) - √ (n-2)/√ (n-1) is more accurately 1/n² + 1/(2.n³).
This finishes my mathematical endeavors.
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