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Fast Series and $log 2$ irrationality
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In a paper a new formula for the zeta function, I found the formula
$$1-sum _k=0^infty frac4 i^(k-1) k left(2 sqrt2-3right)^k (-1)^leftlfloor frack2rightrfloor left(leftlfloor frack2rightrfloor +1right) Gamma left(2 leftlfloor frack-32rightrfloor +5right) Gamma left(2 leftlfloor frack2rightrfloor +1right)left(2 sqrt2+3right) Gamma (k+1) Gamma (k+3)\=log 2$$
I realize using $k=10$ gives $12$ digits correct that seems a fast series in comparison with the Taylor series, is it possible to use it to show the irrationality of $log2$?
calculus
edited Jul 25 at 12:02
Daniel Buck
2,2941623
asked Jul 25 at 11:10
Clerk
548
add a comment |Â
up vote
0
down vote
favorite
In a paper a new formula for the zeta function, I found the formula
$$1-sum _k=0^infty frac4 i^(k-1) k left(2 sqrt2-3right)^k (-1)^leftlfloor frack2rightrfloor left(leftlfloor frack2rightrfloor +1right) Gamma left(2 leftlfloor frack-32rightrfloor +5right) Gamma left(2 leftlfloor frack2rightrfloor +1right)left(2 sqrt2+3right) Gamma (k+1) Gamma (k+3)\=log 2$$
I realize using $k=10$ gives $12$ digits correct that seems a fast series in comparison with the Taylor series, is it possible to use it to show the irrationality of $log2$?
calculus
edited Jul 25 at 12:02
Daniel Buck
2,2941623
asked Jul 25 at 11:10
Clerk
548
Do you mean $log $ base $10$ or base $e$? Either way, to use a series to show irrationality generally requires a very good handle on the error term for the partial sums (trusting that the partial sums are rational). And there are easier proofs of irrationality (in either case).
– lulu
Jul 25 at 11:18
2
And to prove irrationality, we would probably want a series of rationals, but maybe this is not, since it has $sqrt2$ in there?
– GEdgar
Jul 25 at 11:31
Edgar is right, although I suppose it's about accelerating the speed of convergence to demonstrate irrationality
– Clerk
Jul 25 at 20:45
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
In a paper a new formula for the zeta function, I found the formula
$$1-sum _k=0^infty frac4 i^(k-1) k left(2 sqrt2-3right)^k (-1)^leftlfloor frack2rightrfloor left(leftlfloor frack2rightrfloor +1right) Gamma left(2 leftlfloor frack-32rightrfloor +5right) Gamma left(2 leftlfloor frack2rightrfloor +1right)left(2 sqrt2+3right) Gamma (k+1) Gamma (k+3)\=log 2$$
I realize using $k=10$ gives $12$ digits correct that seems a fast series in comparison with the Taylor series, is it possible to use it to show the irrationality of $log2$?
calculus
edited Jul 25 at 12:02
Daniel Buck
2,2941623
asked Jul 25 at 11:10
Clerk
548
In a paper a new formula for the zeta function, I found the formula
$$1-sum _k=0^infty frac4 i^(k-1) k left(2 sqrt2-3right)^k (-1)^leftlfloor frack2rightrfloor left(leftlfloor frack2rightrfloor +1right) Gamma left(2 leftlfloor frack-32rightrfloor +5right) Gamma left(2 leftlfloor frack2rightrfloor +1right)left(2 sqrt2+3right) Gamma (k+1) Gamma (k+3)\=log 2$$
I realize using $k=10$ gives $12$ digits correct that seems a fast series in comparison with the Taylor series, is it possible to use it to show the irrationality of $log2$?
calculus
edited Jul 25 at 12:02
Daniel Buck
2,2941623
asked Jul 25 at 11:10
Clerk
548
edited Jul 25 at 12:02
Daniel Buck
2,2941623
edited Jul 25 at 12:02
Daniel Buck
2,2941623
edited Jul 25 at 12:02
Daniel Buck
2,2941623
2,2941623
asked Jul 25 at 11:10
Clerk
548
asked Jul 25 at 11:10
Clerk
548
asked Jul 25 at 11:10
Clerk
548
548
Do you mean $log $ base $10$ or base $e$? Either way, to use a series to show irrationality generally requires a very good handle on the error term for the partial sums (trusting that the partial sums are rational). And there are easier proofs of irrationality (in either case).
– lulu
Jul 25 at 11:18
2
And to prove irrationality, we would probably want a series of rationals, but maybe this is not, since it has $sqrt2$ in there?
– GEdgar
Jul 25 at 11:31
Edgar is right, although I suppose it's about accelerating the speed of convergence to demonstrate irrationality
– Clerk
Jul 25 at 20:45
add a comment |Â
Do you mean $log $ base $10$ or base $e$? Either way, to use a series to show irrationality generally requires a very good handle on the error term for the partial sums (trusting that the partial sums are rational). And there are easier proofs of irrationality (in either case).
– lulu
Jul 25 at 11:18
2
And to prove irrationality, we would probably want a series of rationals, but maybe this is not, since it has $sqrt2$ in there?
– GEdgar
Jul 25 at 11:31
Edgar is right, although I suppose it's about accelerating the speed of convergence to demonstrate irrationality
– Clerk
Jul 25 at 20:45
Do you mean $log $ base $10$ or base $e$? Either way, to use a series to show irrationality generally requires a very good handle on the error term for the partial sums (trusting that the partial sums are rational). And there are easier proofs of irrationality (in either case).
– lulu
Jul 25 at 11:18
Do you mean $log $ base $10$ or base $e$? Either way, to use a series to show irrationality generally requires a very good handle on the error term for the partial sums (trusting that the partial sums are rational). And there are easier proofs of irrationality (in either case).
– lulu
Jul 25 at 11:18
2
2
And to prove irrationality, we would probably want a series of rationals, but maybe this is not, since it has $sqrt2$ in there?
– GEdgar
Jul 25 at 11:31
And to prove irrationality, we would probably want a series of rationals, but maybe this is not, since it has $sqrt2$ in there?
– GEdgar
Jul 25 at 11:31
Edgar is right, although I suppose it's about accelerating the speed of convergence to demonstrate irrationality
– Clerk
Jul 25 at 20:45
Edgar is right, although I suppose it's about accelerating the speed of convergence to demonstrate irrationality
– Clerk
Jul 25 at 20:45
add a comment |Â
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Do you mean $log $ base $10$ or base $e$? Either way, to use a series to show irrationality generally requires a very good handle on the error term for the partial sums (trusting that the partial sums are rational). And there are easier proofs of irrationality (in either case).
– lulu
Jul 25 at 11:18
2
And to prove irrationality, we would probably want a series of rationals, but maybe this is not, since it has $sqrt2$ in there?
– GEdgar
Jul 25 at 11:31
Edgar is right, although I suppose it's about accelerating the speed of convergence to demonstrate irrationality
– Clerk
Jul 25 at 20:45