Are these upper and lower bounds for $frac{x!}{leftlfloor{x}rightrfloor!}$ useful? If so, are they already...
$begingroup$
Truncating the infinite series for the derivative of the Digamma function
$$
psi'(x) = sum_{n=0}^inftyfrac{1}{(x + n)^2}
$$
after $m-1$ terms, where $m$ is a positive integer (the case $m=2$ answered the question How do we prove that $(x-1)!leq{(frac{x}{2})^{x-1}}$?), finding upper and lower bounds for the remainder, and integrating twice between the limits $2$ and $2+x$ (at least, I think that's what I did, but it was a long slog, and my notation has changed several times since then), one arrives at the inequalities
$$
left(frac{m+1+x}{m+1}right)^{m+1+x}
!!! < frac{e^x(m+x)!}{e^{(H_m-gamma)x}m!} <
left(frac{m+x}{m}right)^{m+x}
quad (x > 0; m = 1, 2, 3, ldots).
$$
This seems most useful (if useful at all!) for smallish $x$. Replacing $m+x$ by $x$ and $m$ by $leftlfloor{x}rightrfloor$, we get
$$
left(frac{x+1}{leftlfloor{x}rightrfloor+1}right)^{x+1}
!!! <
frac{e^{x-leftlfloor{x}rightrfloor}}{e^{(H_m-gamma)(x-leftlfloor{x}rightrfloor)}}
cdot frac{x!}{leftlfloor{x}rightrfloor!}
<
left(frac{x}{leftlfloor{x}rightrfloor}right)^x
quad(x > 1, x notin mathbb{N}).
$$
This seems to give sharper bounds than the following simple exact form of Stirling's approximation:
$$
sqrt{2pi}n^{n+frac{1}{2}}e^{-n} leqslant n! leqslant en^{n+frac{1}{2}}e^{-n}.
$$
On the other hand, it seems to be generally inferior to the full version of Robbins's bounds:
$$
sqrt{2pi}n^{n+frac{1}{2}}e^{-n}e^{frac{1}{12n+1}} < n! < sqrt{2pi}n^{n+frac{1}{2}}e^{-n}e^{frac{1}{12n}}.
$$
For small values of $x - leftlfloor{x}rightrfloor$, my formula does sometimes give better results. For example, $7.04! bumpeq 5463.7647$, and in this case my formula gives the strict bounds $(5463.7292, 5463.8071)$, whereas Robbins's formula gives $(5463.0514, 5463.8080)$, and the simplified version of his formula gives the distinctly worse estimates $(5399.5135, 5855.4353)$.
Might my horrid formula therefore have some actual use? If so, has it been published already? Does it have a less nasty proof than the one I've sketched?
calculus inequality reference-request alternative-proof
asked Jan 7 at 19:41
Calum GilhooleyCalum Gilhooley
5,119730
$endgroup$
add a comment |
$begingroup$
Truncating the infinite series for the derivative of the Digamma function
$$
psi'(x) = sum_{n=0}^inftyfrac{1}{(x + n)^2}
$$
after $m-1$ terms, where $m$ is a positive integer (the case $m=2$ answered the question How do we prove that $(x-1)!leq{(frac{x}{2})^{x-1}}$?), finding upper and lower bounds for the remainder, and integrating twice between the limits $2$ and $2+x$ (at least, I think that's what I did, but it was a long slog, and my notation has changed several times since then), one arrives at the inequalities
$$
left(frac{m+1+x}{m+1}right)^{m+1+x}
!!! < frac{e^x(m+x)!}{e^{(H_m-gamma)x}m!} <
left(frac{m+x}{m}right)^{m+x}
quad (x > 0; m = 1, 2, 3, ldots).
$$
This seems most useful (if useful at all!) for smallish $x$. Replacing $m+x$ by $x$ and $m$ by $leftlfloor{x}rightrfloor$, we get
$$
left(frac{x+1}{leftlfloor{x}rightrfloor+1}right)^{x+1}
!!! <
frac{e^{x-leftlfloor{x}rightrfloor}}{e^{(H_m-gamma)(x-leftlfloor{x}rightrfloor)}}
cdot frac{x!}{leftlfloor{x}rightrfloor!}
<
left(frac{x}{leftlfloor{x}rightrfloor}right)^x
quad(x > 1, x notin mathbb{N}).
$$
This seems to give sharper bounds than the following simple exact form of Stirling's approximation:
$$
sqrt{2pi}n^{n+frac{1}{2}}e^{-n} leqslant n! leqslant en^{n+frac{1}{2}}e^{-n}.
$$
On the other hand, it seems to be generally inferior to the full version of Robbins's bounds:
$$
sqrt{2pi}n^{n+frac{1}{2}}e^{-n}e^{frac{1}{12n+1}} < n! < sqrt{2pi}n^{n+frac{1}{2}}e^{-n}e^{frac{1}{12n}}.
$$
For small values of $x - leftlfloor{x}rightrfloor$, my formula does sometimes give better results. For example, $7.04! bumpeq 5463.7647$, and in this case my formula gives the strict bounds $(5463.7292, 5463.8071)$, whereas Robbins's formula gives $(5463.0514, 5463.8080)$, and the simplified version of his formula gives the distinctly worse estimates $(5399.5135, 5855.4353)$.
Might my horrid formula therefore have some actual use? If so, has it been published already? Does it have a less nasty proof than the one I've sketched?
calculus inequality reference-request alternative-proof
asked Jan 7 at 19:41
Calum GilhooleyCalum Gilhooley
5,119730
$endgroup$
add a comment |
$begingroup$
Truncating the infinite series for the derivative of the Digamma function
$$
psi'(x) = sum_{n=0}^inftyfrac{1}{(x + n)^2}
$$
after $m-1$ terms, where $m$ is a positive integer (the case $m=2$ answered the question How do we prove that $(x-1)!leq{(frac{x}{2})^{x-1}}$?), finding upper and lower bounds for the remainder, and integrating twice between the limits $2$ and $2+x$ (at least, I think that's what I did, but it was a long slog, and my notation has changed several times since then), one arrives at the inequalities
$$
left(frac{m+1+x}{m+1}right)^{m+1+x}
!!! < frac{e^x(m+x)!}{e^{(H_m-gamma)x}m!} <
left(frac{m+x}{m}right)^{m+x}
quad (x > 0; m = 1, 2, 3, ldots).
$$
This seems most useful (if useful at all!) for smallish $x$. Replacing $m+x$ by $x$ and $m$ by $leftlfloor{x}rightrfloor$, we get
$$
left(frac{x+1}{leftlfloor{x}rightrfloor+1}right)^{x+1}
!!! <
frac{e^{x-leftlfloor{x}rightrfloor}}{e^{(H_m-gamma)(x-leftlfloor{x}rightrfloor)}}
cdot frac{x!}{leftlfloor{x}rightrfloor!}
<
left(frac{x}{leftlfloor{x}rightrfloor}right)^x
quad(x > 1, x notin mathbb{N}).
$$
This seems to give sharper bounds than the following simple exact form of Stirling's approximation:
$$
sqrt{2pi}n^{n+frac{1}{2}}e^{-n} leqslant n! leqslant en^{n+frac{1}{2}}e^{-n}.
$$
On the other hand, it seems to be generally inferior to the full version of Robbins's bounds:
$$
sqrt{2pi}n^{n+frac{1}{2}}e^{-n}e^{frac{1}{12n+1}} < n! < sqrt{2pi}n^{n+frac{1}{2}}e^{-n}e^{frac{1}{12n}}.
$$
For small values of $x - leftlfloor{x}rightrfloor$, my formula does sometimes give better results. For example, $7.04! bumpeq 5463.7647$, and in this case my formula gives the strict bounds $(5463.7292, 5463.8071)$, whereas Robbins's formula gives $(5463.0514, 5463.8080)$, and the simplified version of his formula gives the distinctly worse estimates $(5399.5135, 5855.4353)$.
Might my horrid formula therefore have some actual use? If so, has it been published already? Does it have a less nasty proof than the one I've sketched?
calculus inequality reference-request alternative-proof
asked Jan 7 at 19:41
Calum GilhooleyCalum Gilhooley
5,119730
$endgroup$
Truncating the infinite series for the derivative of the Digamma function
$$
psi'(x) = sum_{n=0}^inftyfrac{1}{(x + n)^2}
$$
after $m-1$ terms, where $m$ is a positive integer (the case $m=2$ answered the question How do we prove that $(x-1)!leq{(frac{x}{2})^{x-1}}$?), finding upper and lower bounds for the remainder, and integrating twice between the limits $2$ and $2+x$ (at least, I think that's what I did, but it was a long slog, and my notation has changed several times since then), one arrives at the inequalities
$$
left(frac{m+1+x}{m+1}right)^{m+1+x}
!!! < frac{e^x(m+x)!}{e^{(H_m-gamma)x}m!} <
left(frac{m+x}{m}right)^{m+x}
quad (x > 0; m = 1, 2, 3, ldots).
$$
This seems most useful (if useful at all!) for smallish $x$. Replacing $m+x$ by $x$ and $m$ by $leftlfloor{x}rightrfloor$, we get
$$
left(frac{x+1}{leftlfloor{x}rightrfloor+1}right)^{x+1}
!!! <
frac{e^{x-leftlfloor{x}rightrfloor}}{e^{(H_m-gamma)(x-leftlfloor{x}rightrfloor)}}
cdot frac{x!}{leftlfloor{x}rightrfloor!}
<
left(frac{x}{leftlfloor{x}rightrfloor}right)^x
quad(x > 1, x notin mathbb{N}).
$$
This seems to give sharper bounds than the following simple exact form of Stirling's approximation:
$$
sqrt{2pi}n^{n+frac{1}{2}}e^{-n} leqslant n! leqslant en^{n+frac{1}{2}}e^{-n}.
$$
On the other hand, it seems to be generally inferior to the full version of Robbins's bounds:
$$
sqrt{2pi}n^{n+frac{1}{2}}e^{-n}e^{frac{1}{12n+1}} < n! < sqrt{2pi}n^{n+frac{1}{2}}e^{-n}e^{frac{1}{12n}}.
$$
For small values of $x - leftlfloor{x}rightrfloor$, my formula does sometimes give better results. For example, $7.04! bumpeq 5463.7647$, and in this case my formula gives the strict bounds $(5463.7292, 5463.8071)$, whereas Robbins's formula gives $(5463.0514, 5463.8080)$, and the simplified version of his formula gives the distinctly worse estimates $(5399.5135, 5855.4353)$.
Might my horrid formula therefore have some actual use? If so, has it been published already? Does it have a less nasty proof than the one I've sketched?
calculus inequality reference-request alternative-proof
calculus inequality reference-request alternative-proof
asked Jan 7 at 19:41
Calum GilhooleyCalum Gilhooley
5,119730
asked Jan 7 at 19:41
Calum GilhooleyCalum Gilhooley
5,119730
asked Jan 7 at 19:41
Calum GilhooleyCalum Gilhooley
5,119730
asked Jan 7 at 19:41
Calum GilhooleyCalum Gilhooley
5,119730
asked Jan 7 at 19:41
Calum GilhooleyCalum Gilhooley
5,119730
5,119730
add a comment |
add a comment |
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