Mixing
There is at most one zero of function in given interval.
$begingroup$
Let the function $f_{n}:mathbb{R}rightarrow mathbb{R_{ge 0}}$ be defined as
$$
f_{n}(x)=sum_{i,j=1}^{n}frac{(-1)^{i+j}cos(ln frac{i}{j})}{(ij)^{x}}quad forall ninBbb N
$$
and let also
$$
F_{n}(x)=left(sqrt{f_{n}(x)}-frac{1}{(n+1)^{x}}right)^{2} quad forall ninBbb N.
$$
Is it true that for every $n$ equation $F_{n}'(x)=0$ has at most one solution in the interval $xin(0,1)$?
My idea was to prove that difference between consecutive zeros of given derivative is $ge 1$
Also i wrote given functions in GeoGebra program for some different $n$ and everything seems to be ok, but I don't know how to check it in general.
Regards.
real-analysis summation special-functions
edited Jan 27 at 10:29
Daniele Tampieri
2,5772922
asked Jan 27 at 10:22
mkultramkultra
988
$endgroup$
add a comment |
$begingroup$
Let the function $f_{n}:mathbb{R}rightarrow mathbb{R_{ge 0}}$ be defined as
$$
f_{n}(x)=sum_{i,j=1}^{n}frac{(-1)^{i+j}cos(ln frac{i}{j})}{(ij)^{x}}quad forall ninBbb N
$$
and let also
$$
F_{n}(x)=left(sqrt{f_{n}(x)}-frac{1}{(n+1)^{x}}right)^{2} quad forall ninBbb N.
$$
Is it true that for every $n$ equation $F_{n}'(x)=0$ has at most one solution in the interval $xin(0,1)$?
My idea was to prove that difference between consecutive zeros of given derivative is $ge 1$
Also i wrote given functions in GeoGebra program for some different $n$ and everything seems to be ok, but I don't know how to check it in general.
Regards.
real-analysis summation special-functions
edited Jan 27 at 10:29
Daniele Tampieri
2,5772922
asked Jan 27 at 10:22
mkultramkultra
988
$endgroup$
$begingroup$
$F_n$ will have at most one zero if $sqrt{f_n(x)} - (n+1)^{-x}$ is monotonic. It may also help to notice that the terms in the sum for $f_n(x)$ are symmetric in $i$ and $j$. I.e., exchanging the values of $i$ and $j$ does not affect the term.
$endgroup$
– Paul Sinclair
Jan 27 at 16:58
add a comment |
$begingroup$
Let the function $f_{n}:mathbb{R}rightarrow mathbb{R_{ge 0}}$ be defined as
$$
f_{n}(x)=sum_{i,j=1}^{n}frac{(-1)^{i+j}cos(ln frac{i}{j})}{(ij)^{x}}quad forall ninBbb N
$$
and let also
$$
F_{n}(x)=left(sqrt{f_{n}(x)}-frac{1}{(n+1)^{x}}right)^{2} quad forall ninBbb N.
$$
Is it true that for every $n$ equation $F_{n}'(x)=0$ has at most one solution in the interval $xin(0,1)$?
My idea was to prove that difference between consecutive zeros of given derivative is $ge 1$
Also i wrote given functions in GeoGebra program for some different $n$ and everything seems to be ok, but I don't know how to check it in general.
Regards.
real-analysis summation special-functions
edited Jan 27 at 10:29
Daniele Tampieri
2,5772922
asked Jan 27 at 10:22
mkultramkultra
988
$endgroup$
Let the function $f_{n}:mathbb{R}rightarrow mathbb{R_{ge 0}}$ be defined as
$$
f_{n}(x)=sum_{i,j=1}^{n}frac{(-1)^{i+j}cos(ln frac{i}{j})}{(ij)^{x}}quad forall ninBbb N
$$
and let also
$$
F_{n}(x)=left(sqrt{f_{n}(x)}-frac{1}{(n+1)^{x}}right)^{2} quad forall ninBbb N.
$$
Is it true that for every $n$ equation $F_{n}'(x)=0$ has at most one solution in the interval $xin(0,1)$?
My idea was to prove that difference between consecutive zeros of given derivative is $ge 1$
Also i wrote given functions in GeoGebra program for some different $n$ and everything seems to be ok, but I don't know how to check it in general.
Regards.
real-analysis summation special-functions
real-analysis summation special-functions
edited Jan 27 at 10:29
Daniele Tampieri
2,5772922
asked Jan 27 at 10:22
mkultramkultra
988
edited Jan 27 at 10:29
Daniele Tampieri
2,5772922
asked Jan 27 at 10:22
mkultramkultra
988
edited Jan 27 at 10:29
Daniele Tampieri
2,5772922
edited Jan 27 at 10:29
Daniele Tampieri
2,5772922
edited Jan 27 at 10:29
Daniele Tampieri
2,5772922
2,5772922
asked Jan 27 at 10:22
mkultramkultra
988
asked Jan 27 at 10:22
mkultramkultra
988
asked Jan 27 at 10:22
mkultramkultra
988
988
$begingroup$
$F_n$ will have at most one zero if $sqrt{f_n(x)} - (n+1)^{-x}$ is monotonic. It may also help to notice that the terms in the sum for $f_n(x)$ are symmetric in $i$ and $j$. I.e., exchanging the values of $i$ and $j$ does not affect the term.
$endgroup$
– Paul Sinclair
Jan 27 at 16:58
add a comment |
$begingroup$
$F_n$ will have at most one zero if $sqrt{f_n(x)} - (n+1)^{-x}$ is monotonic. It may also help to notice that the terms in the sum for $f_n(x)$ are symmetric in $i$ and $j$. I.e., exchanging the values of $i$ and $j$ does not affect the term.
$endgroup$
– Paul Sinclair
Jan 27 at 16:58
$begingroup$
$F_n$ will have at most one zero if $sqrt{f_n(x)} - (n+1)^{-x}$ is monotonic. It may also help to notice that the terms in the sum for $f_n(x)$ are symmetric in $i$ and $j$. I.e., exchanging the values of $i$ and $j$ does not affect the term.
$endgroup$
– Paul Sinclair
Jan 27 at 16:58
$begingroup$
$F_n$ will have at most one zero if $sqrt{f_n(x)} - (n+1)^{-x}$ is monotonic. It may also help to notice that the terms in the sum for $f_n(x)$ are symmetric in $i$ and $j$. I.e., exchanging the values of $i$ and $j$ does not affect the term.
$endgroup$
– Paul Sinclair
Jan 27 at 16:58
add a comment |
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$begingroup$
$F_n$ will have at most one zero if $sqrt{f_n(x)} - (n+1)^{-x}$ is monotonic. It may also help to notice that the terms in the sum for $f_n(x)$ are symmetric in $i$ and $j$. I.e., exchanging the values of $i$ and $j$ does not affect the term.
$endgroup$
– Paul Sinclair
Jan 27 at 16:58