Reputation
Top tag
Next privilege 75 Rep.
Set bounties
Badges
1 16
Impact
~8k people reached

  • 0 posts edited
  • 2 helpful flags
  • 8 votes cast
1d
revised Seeking proof for the formula relating Pi with its convergents
added 8 characters in body
1d
revised Seeking proof for the formula relating Pi with its convergents
deleted 299 characters in body
1d
revised Seeking proof for the formula relating Pi with its convergents
deleted 325 characters in body
Feb
6
comment Seeking proof for the formula relating Pi with its convergents
@Jaume Oliver Lafont - so perhaps now one could produce two distinct families of parameterization: one for Pi and the differences between Pi and its convergents and another for log(2) and the differences between log(2) and its convergents :-)
Feb
6
comment Seeking proof for the formula relating Pi with its convergents
@Jaume Oliver Lafont - thanks, fascinating ...
Feb
6
comment Seeking proof for the formula relating Pi with its convergents
@Jaume Oliver Lafont - it is interesting that depending on the growth of the polynomial x degree in the numerator (while denominator stays to be the same "1+x^2") the result is changing from "Pi" to "log(2)" and then to "+/- (Pi - p/q)" ...
Feb
5
comment Is there an integral that proves $\pi > 333/106$?
The 1st case in the above answer fits i=-1, j=2, k=1, l=4 for parametric formula $$ (-1)^n\cdot(\pi - \text{A002485}(n)/\text{A002486}(n)) =(|i|\cdot2^j)^{-1} \int_0^1 \big(x^l(1-x)^{2(j+2)}(k+(i+k)x^2)\big)/(1+x^2)\; dx $$ The second case fits parameters: $$ i=-1, j=2, k=1, l=8 $$ The 3rd case fits at least the following 2 combinations $$ i=-1, j=-1, k=1, l=1 $$ $$ i=-2, j=-1, k=2, l=1 $$ $$ (abs(-1)*(-1)^2)^(-1)*Int((x^1*(1-x)^(2*(-1+2))*(1+(-1+1)*x^2))/(1+x^2),x=0...1)‌​=(Pi -3)/2 $$ $$ (abs(-2)*(-1)^2)^(-1)*Int((x^1*(1-x)^(2*(-1+2))*(2+(-2+2)*x^2))/(1+x^2),x=0...1)‌​=(Pi -3)/2 $$
Feb
3
revised Is there an integral that proves $\pi > 333/106$?
added 54 characters in body
Feb
3
revised Seeking proof for the formula relating Pi with its convergents
added 54 characters in body
Feb
1
revised Is there an integral that proves $\pi > 333/106$?
added 722 characters in body
Feb
1
revised Seeking proof for the formula relating Pi with its convergents
deleted 140 characters in body
Feb
1
comment Seeking proof for the formula relating Pi with its convergents
@Jaume Oliver Lafont - that will be the case where i=-1, j=-2, k=1, l=0 Should there be infinite number of such cases?
Feb
1
comment Seeking proof for the formula relating Pi with its convergents
@Jaume Oliver Lafont - On another hand, if your guess is correct, then why not to assume that with some parameter values RHS of this parametric identity satisfy the case when the rational fraction on the LHS is equal to zero? If so, then it becomes an identity for Pi ... BTW, if I did not make a mistake, RHS could be reduced (after performing integration) to: (abs(i)*2^j)^(-1)*Gamma(2*j+5)*((k+i)*Gamma(l+3)*HypergeometricPFQ(1,l/2+3/2,l/2‌​+2;j+l/2+4,j+l/2+9/2;-1)/Gamma(2*j+l+8)+k*Gamma(l+1)*HypergeometricPFQ(1,l/2+1/2,‌​l/2+1;j+l/2+3, j+l/2+7/2;-1)/Gamma(2*j+l+6))
Feb
1
comment A series related to $\pi\approx 2\sqrt{1+\sqrt{2}}$
I would appreciate if one of moderators would convert this to a comment. Perhaps the original question could be expanded and other relatively simple $\pi$ approximations should be considered? For example, few years ago I found that $$\pi \approx \sqrt{4\cdot e^{1} - 1}.$$
Jan
31
revised Seeking proof for the formula relating Pi with its convergents
added 795 characters in body
Jan
31
revised Is there an integral that proves $\pi > 333/106$?
added 264 characters in body
Jan
31
revised Seeking proof for the formula relating Pi with its convergents
added 268 characters in body
Jan
31
revised Is there an integral that proves $\pi > 333/106$?
added 1 character in body
Jan
31
revised Seeking proof for the formula relating Pi with its convergents
added 11 characters in body
Jan
31
comment Seeking proof for the formula relating Pi with its convergents
@Jaume Oliver Lafont - if your guess is correct, then {i,j,k,l} on the right hand side are not integer functions of "n" (in the sense of "n" being an index of A002485 and A002486) on the left hand side? - btw, in processed cases it is seen that values of "i" are growing as fraction approaches Pi - it makes sense, because when LHS is approaching 0, RHS should also do so, and i --> "infinity" does this. It is also interesting that, as Thomas Baruchel's calculations show, even for each particular fraction, being considered, there are infinite number of 4-tuple {i,j,k,l} solutions - puzzling.