Alex
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 Sep24 awarded Autobiographer Sep23 awarded Nice Answer Jul24 revised Is there an integral that proves $\pi > 333/106$? added 1853 characters in body Jul22 comment Seeking proof for the formula relating Pi with its convergents Obviously parameters in the formula somehow depend on "n". The most straight forward dependency on "n" is observed in what you call "m'" (and I call "p") ;{2,3,4,4,6,6} ... It appears that when "n" -> infinity - then the integral should come to 0 ... Jul22 comment Seeking proof for the formula relating Pi with its convergents Pi - 103993/33102 -> a:=124360;b:=77159;p:=6;c:=2;int((x^(c+2*p)*(1-x)^(2*p)*(a+bx^2))/((a-b)*2^(p-2‌​‌​)*((-1)^(c)*(1+x^2))),x=0...1) 355/113 - Pi -> a:=25;b:=816;p:=4;c:=0;int((x^(c+2*p)*(1-x)^(2*p)*(a+bx^2))/((a-b)*2^(p-2)*((-1‌​‌​)^(c)*(1+x^2))),x=0...1) Pi - 333/106 -> a:=197;b:=462;p:=3;c:=-1;int((x^(c+2*p)*(1-x)^(2*p)(a+bx^2))/((a-b)*2^(p-2)*((-‌​‌​1)^(c)*(1+x^2))),x=0...1) 22/7 - Pi -> a:=1;b:=0;p:=2;c:=0;int((x^(c+2*p)*(1-x)^(2*p)*(a+bx^2))/((a-b)*2^(p-2)*((-1)‌​‌​^(c)*(1+x^2))),x=0...1) Jul20 revised Question on linear partial difference equation with three independent variables $n$, $m$, $k$. deleted 4 characters in body Jul20 asked Question on linear partial difference equation with three independent variables $n$, $m$, $k$. Jul18 comment Seeking proof for the formula relating Pi with its convergents @(Matt B.) In the next two comments I list parameters in your formula for all cases (I replaced alpha by a, bets by b, epsilon by c, m' by p) 104348/33215 - Pi -> a:=1349;b:=-1060;p:=6;c:=0;int((x^(c+2*p)*(1-x)^(2*p)*(a+b*x^2))/((a-b)*2^(p-2)*‌​((-1)^(c)*(1+x^2))),x=0...1) Jul17 awarded Benefactor Jul17 accepted Seeking proof for the formula relating Pi with its convergents Jul17 awarded Self-Learner Jul16 comment Permutation identities similar to $(7901234568 / 9876543210) \cdot 1234567890 = 0987654312$ @Gerry Myerson - systematic search ... That is what was done programmatically ... Jul16 comment Permutation identities similar to $(7901234568 / 9876543210) \cdot 1234567890 = 0987654312$ @Gerry Myerson Thanks - I think that you have proved that there are at least two pairs - yes ? If so, is there any way to evaluate how many (more than 2) such pairs exist for the given base with n>2 ? Jul16 awarded Nice Question Jul15 comment Why this recursively defined sequence of real numbers converges to -Pi? @nbubis - thanks for the info - it worked out that way anyway. Jul15 comment Why this recursively defined sequence of real numbers converges to -Pi? @nbubis - I was hoping though that you will return to address additional questions (see my comments made on July 6 and July 9 ) towards which I really issued the bounty :-) Jul15 comment Why this recursively defined sequence of real numbers converges to -Pi? @nbubis But it went to you automatically (from how I understand the rules) - correct ? Jul15 awarded Revival Jul15 revised Permutation identities similar to $(7901234568 / 9876543210) \cdot 1234567890 = 0987654312$ added 3 characters in body Jul15 revised Permutation identities similar to $(7901234568 / 9876543210) \cdot 1234567890 = 0987654312$ added 51 characters in body