Mixing
What is the minimum value of $f_infty=frac{x}{sqrt{x-sqrt[3]{x-sqrt[4]{x-cdots}}}}$?
$begingroup$
In a similar vein to What is the maximum value of this nested radical?, I'd like to share a similar nested radical, but this time with changing fractional powers.
What is the minimum value of $$f_infty=frac{x}{sqrt{x-sqrt[3]{x-sqrt[4]{x-cdots}}}}$$ where the radicals go up by one each time?
- Here is a plot of $f_{19}$. We can see that as $xto 1^+$, $min f_{19}to 1.7186$ which is strange as the denominator can only take the binary values $0$ or $1$ at $x=1$. The curve is monotonically increasing from $1$ onwards, which is expected as the numerator dominates.
Actually, a simulation in PARI/GP up to $f_{100}$ yields a minimum value of around $1.718565$, which is somewhat close to $e-1$, although I strongly doubt that it will ever reach that value.
Note that $f_k$ is defined in $(1,infty)$ for all positive integers $k$, but the curve swings wildly for $(-infty,1)$. It is, of course, not a good idea to differentiate $f_infty$ directly, but unfortunately we can't take $x$ and $sqrt{x-sqrt[3]{x-sqrt[4]{x-cdots}}}$ separately as both are increasing.
Another interesting question: Why is the minimum value of $f_k$ for large $k$ not equal to the expected $0,1$ or $pminfty$? Is it possible to manipulate $f_infty$ so that L'Hopital can be used to find the value of $1.718cdots$?
Related are
Evaluating the limit of $sqrt[2]{2+sqrt[3]{2+sqrt[4]{2+cdots+sqrt[n]{2}}}}$ when $ntoinfty$
Find $sqrt{4+sqrt[3]{4+sqrt[4]{4+sqrt[5]{4+cdots}}}}$
but neither have been solved as of now.
limits functions recursion maxima-minima nested-radicals
asked Jan 19 at 21:14
TheSimpliFireTheSimpliFire
12.6k62461
$endgroup$
add a comment |
$begingroup$
In a similar vein to What is the maximum value of this nested radical?, I'd like to share a similar nested radical, but this time with changing fractional powers.
What is the minimum value of $$f_infty=frac{x}{sqrt{x-sqrt[3]{x-sqrt[4]{x-cdots}}}}$$ where the radicals go up by one each time?
- Here is a plot of $f_{19}$. We can see that as $xto 1^+$, $min f_{19}to 1.7186$ which is strange as the denominator can only take the binary values $0$ or $1$ at $x=1$. The curve is monotonically increasing from $1$ onwards, which is expected as the numerator dominates.
Actually, a simulation in PARI/GP up to $f_{100}$ yields a minimum value of around $1.718565$, which is somewhat close to $e-1$, although I strongly doubt that it will ever reach that value.
Note that $f_k$ is defined in $(1,infty)$ for all positive integers $k$, but the curve swings wildly for $(-infty,1)$. It is, of course, not a good idea to differentiate $f_infty$ directly, but unfortunately we can't take $x$ and $sqrt{x-sqrt[3]{x-sqrt[4]{x-cdots}}}$ separately as both are increasing.
Another interesting question: Why is the minimum value of $f_k$ for large $k$ not equal to the expected $0,1$ or $pminfty$? Is it possible to manipulate $f_infty$ so that L'Hopital can be used to find the value of $1.718cdots$?
Related are
Evaluating the limit of $sqrt[2]{2+sqrt[3]{2+sqrt[4]{2+cdots+sqrt[n]{2}}}}$ when $ntoinfty$
Find $sqrt{4+sqrt[3]{4+sqrt[4]{4+sqrt[5]{4+cdots}}}}$
but neither have been solved as of now.
limits functions recursion maxima-minima nested-radicals
asked Jan 19 at 21:14
TheSimpliFireTheSimpliFire
12.6k62461
$endgroup$
1
$begingroup$
L'Hospital cannot be used since for $xrightarrow 1$, the numerator tends to $1$.
$endgroup$
– Peter
Jan 19 at 21:23
$begingroup$
@Peter I know, I was wondering if we can write $f_infty$ in a different way so that both numerator and denominator are $0$ or $infty$.
$endgroup$
– TheSimpliFire
Jan 19 at 21:26
1
$begingroup$
The denominator tends to $$0.581880523059785627121cdots$$ for $xrightarrow 1$
$endgroup$
– Peter
Jan 19 at 21:29
2
$begingroup$
I sincerely doubt the minimum has a nice expression. The increasing roots are going to ensure that.
$endgroup$
– Brevan Ellefsen
Jan 19 at 21:31
add a comment |
$begingroup$
In a similar vein to What is the maximum value of this nested radical?, I'd like to share a similar nested radical, but this time with changing fractional powers.
What is the minimum value of $$f_infty=frac{x}{sqrt{x-sqrt[3]{x-sqrt[4]{x-cdots}}}}$$ where the radicals go up by one each time?
- Here is a plot of $f_{19}$. We can see that as $xto 1^+$, $min f_{19}to 1.7186$ which is strange as the denominator can only take the binary values $0$ or $1$ at $x=1$. The curve is monotonically increasing from $1$ onwards, which is expected as the numerator dominates.
Actually, a simulation in PARI/GP up to $f_{100}$ yields a minimum value of around $1.718565$, which is somewhat close to $e-1$, although I strongly doubt that it will ever reach that value.
Note that $f_k$ is defined in $(1,infty)$ for all positive integers $k$, but the curve swings wildly for $(-infty,1)$. It is, of course, not a good idea to differentiate $f_infty$ directly, but unfortunately we can't take $x$ and $sqrt{x-sqrt[3]{x-sqrt[4]{x-cdots}}}$ separately as both are increasing.
Another interesting question: Why is the minimum value of $f_k$ for large $k$ not equal to the expected $0,1$ or $pminfty$? Is it possible to manipulate $f_infty$ so that L'Hopital can be used to find the value of $1.718cdots$?
Related are
Evaluating the limit of $sqrt[2]{2+sqrt[3]{2+sqrt[4]{2+cdots+sqrt[n]{2}}}}$ when $ntoinfty$
Find $sqrt{4+sqrt[3]{4+sqrt[4]{4+sqrt[5]{4+cdots}}}}$
but neither have been solved as of now.
limits functions recursion maxima-minima nested-radicals
asked Jan 19 at 21:14
TheSimpliFireTheSimpliFire
12.6k62461
$endgroup$
In a similar vein to What is the maximum value of this nested radical?, I'd like to share a similar nested radical, but this time with changing fractional powers.
What is the minimum value of $$f_infty=frac{x}{sqrt{x-sqrt[3]{x-sqrt[4]{x-cdots}}}}$$ where the radicals go up by one each time?
- Here is a plot of $f_{19}$. We can see that as $xto 1^+$, $min f_{19}to 1.7186$ which is strange as the denominator can only take the binary values $0$ or $1$ at $x=1$. The curve is monotonically increasing from $1$ onwards, which is expected as the numerator dominates.
Actually, a simulation in PARI/GP up to $f_{100}$ yields a minimum value of around $1.718565$, which is somewhat close to $e-1$, although I strongly doubt that it will ever reach that value.
Note that $f_k$ is defined in $(1,infty)$ for all positive integers $k$, but the curve swings wildly for $(-infty,1)$. It is, of course, not a good idea to differentiate $f_infty$ directly, but unfortunately we can't take $x$ and $sqrt{x-sqrt[3]{x-sqrt[4]{x-cdots}}}$ separately as both are increasing.
Another interesting question: Why is the minimum value of $f_k$ for large $k$ not equal to the expected $0,1$ or $pminfty$? Is it possible to manipulate $f_infty$ so that L'Hopital can be used to find the value of $1.718cdots$?
Related are
Evaluating the limit of $sqrt[2]{2+sqrt[3]{2+sqrt[4]{2+cdots+sqrt[n]{2}}}}$ when $ntoinfty$
Find $sqrt{4+sqrt[3]{4+sqrt[4]{4+sqrt[5]{4+cdots}}}}$
but neither have been solved as of now.
limits functions recursion maxima-minima nested-radicals
limits functions recursion maxima-minima nested-radicals
asked Jan 19 at 21:14
TheSimpliFireTheSimpliFire
12.6k62461
asked Jan 19 at 21:14
TheSimpliFireTheSimpliFire
12.6k62461
edited Jan 19 at 21:25
TheSimpliFire
asked Jan 19 at 21:14
TheSimpliFireTheSimpliFire
12.6k62461
asked Jan 19 at 21:14
TheSimpliFireTheSimpliFire
12.6k62461
asked Jan 19 at 21:14
TheSimpliFireTheSimpliFire
12.6k62461
12.6k62461
1
$begingroup$
L'Hospital cannot be used since for $xrightarrow 1$, the numerator tends to $1$.
$endgroup$
– Peter
Jan 19 at 21:23
$begingroup$
@Peter I know, I was wondering if we can write $f_infty$ in a different way so that both numerator and denominator are $0$ or $infty$.
$endgroup$
– TheSimpliFire
Jan 19 at 21:26
1
$begingroup$
The denominator tends to $$0.581880523059785627121cdots$$ for $xrightarrow 1$
$endgroup$
– Peter
Jan 19 at 21:29
2
$begingroup$
I sincerely doubt the minimum has a nice expression. The increasing roots are going to ensure that.
$endgroup$
– Brevan Ellefsen
Jan 19 at 21:31
add a comment |
1
$begingroup$
L'Hospital cannot be used since for $xrightarrow 1$, the numerator tends to $1$.
$endgroup$
– Peter
Jan 19 at 21:23
$begingroup$
@Peter I know, I was wondering if we can write $f_infty$ in a different way so that both numerator and denominator are $0$ or $infty$.
$endgroup$
– TheSimpliFire
Jan 19 at 21:26
1
$begingroup$
The denominator tends to $$0.581880523059785627121cdots$$ for $xrightarrow 1$
$endgroup$
– Peter
Jan 19 at 21:29
2
$begingroup$
I sincerely doubt the minimum has a nice expression. The increasing roots are going to ensure that.
$endgroup$
– Brevan Ellefsen
Jan 19 at 21:31
1
1
$begingroup$
L'Hospital cannot be used since for $xrightarrow 1$, the numerator tends to $1$.
$endgroup$
– Peter
Jan 19 at 21:23
$begingroup$
L'Hospital cannot be used since for $xrightarrow 1$, the numerator tends to $1$.
$endgroup$
– Peter
Jan 19 at 21:23
$begingroup$
@Peter I know, I was wondering if we can write $f_infty$ in a different way so that both numerator and denominator are $0$ or $infty$.
$endgroup$
– TheSimpliFire
Jan 19 at 21:26
$begingroup$
@Peter I know, I was wondering if we can write $f_infty$ in a different way so that both numerator and denominator are $0$ or $infty$.
$endgroup$
– TheSimpliFire
Jan 19 at 21:26
1
1
$begingroup$
The denominator tends to $$0.581880523059785627121cdots$$ for $xrightarrow 1$
$endgroup$
– Peter
Jan 19 at 21:29
$begingroup$
The denominator tends to $$0.581880523059785627121cdots$$ for $xrightarrow 1$
$endgroup$
– Peter
Jan 19 at 21:29
2
2
$begingroup$
I sincerely doubt the minimum has a nice expression. The increasing roots are going to ensure that.
$endgroup$
– Brevan Ellefsen
Jan 19 at 21:31
$begingroup$
I sincerely doubt the minimum has a nice expression. The increasing roots are going to ensure that.
$endgroup$
– Brevan Ellefsen
Jan 19 at 21:31
add a comment |
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1
$begingroup$
L'Hospital cannot be used since for $xrightarrow 1$, the numerator tends to $1$.
$endgroup$
– Peter
Jan 19 at 21:23
$begingroup$
@Peter I know, I was wondering if we can write $f_infty$ in a different way so that both numerator and denominator are $0$ or $infty$.
$endgroup$
– TheSimpliFire
Jan 19 at 21:26
1
$begingroup$
The denominator tends to $$0.581880523059785627121cdots$$ for $xrightarrow 1$
$endgroup$
– Peter
Jan 19 at 21:29
2
$begingroup$
I sincerely doubt the minimum has a nice expression. The increasing roots are going to ensure that.
$endgroup$
– Brevan Ellefsen
Jan 19 at 21:31