Mixing
Find constants $b_i$ such that $sum_{n=1}^{N} a_n (sum_{n=1}^{t} b_i[r_n-(r_n)^2]+s(i))=0$
$begingroup$
I am trying to find the real constants $b_i$ such the $displaystyle sum_{n=1}^{N} a_n left(displaystyle sum_{n=1}^{t} b_i[r_n-(r_n)^2]+s(i))right)=0$ where I am given that $displaystyle sum_{n=1}^{N} a_n r_n=0$ for every natural $N$ and $s(i)$ is certain function of $i$ (I have the explicit form for $s(i)$ as well in terms of a series).
Note that $t$ is at our disposal. I tried to manipulate my expression for $t=3$ to find such constants $b_1,b_2,b_3$ unsuccessfully. I also want to mention that there is no guarantee for the existence of such constants except $b_i=0$ for all $i$.
EDIT: I reviewed my work and I found my original question was wrong. I took the index $N$ in a wrong direction. Now in above question $N$ (say $N=2$) is fixed so that $displaystyle sum_{n=1}^{2} a_n r_n=0$ with $a_1,a_2inmathbb{R}$ and $r_1,r_2in (0,1]$.
Now for fixed but arbitrary $N$ (and corresponding $a_i,r_i$), I am supposed to choose $t$ number of constants $b_i$ (not all zero) making that expression zero.
real-analysis sequences-and-series summation
asked Jan 2 at 22:32
ershersh
294112
$endgroup$
add a comment |
$begingroup$
I am trying to find the real constants $b_i$ such the $displaystyle sum_{n=1}^{N} a_n left(displaystyle sum_{n=1}^{t} b_i[r_n-(r_n)^2]+s(i))right)=0$ where I am given that $displaystyle sum_{n=1}^{N} a_n r_n=0$ for every natural $N$ and $s(i)$ is certain function of $i$ (I have the explicit form for $s(i)$ as well in terms of a series).
Note that $t$ is at our disposal. I tried to manipulate my expression for $t=3$ to find such constants $b_1,b_2,b_3$ unsuccessfully. I also want to mention that there is no guarantee for the existence of such constants except $b_i=0$ for all $i$.
EDIT: I reviewed my work and I found my original question was wrong. I took the index $N$ in a wrong direction. Now in above question $N$ (say $N=2$) is fixed so that $displaystyle sum_{n=1}^{2} a_n r_n=0$ with $a_1,a_2inmathbb{R}$ and $r_1,r_2in (0,1]$.
Now for fixed but arbitrary $N$ (and corresponding $a_i,r_i$), I am supposed to choose $t$ number of constants $b_i$ (not all zero) making that expression zero.
real-analysis sequences-and-series summation
asked Jan 2 at 22:32
ershersh
294112
$endgroup$
3
$begingroup$
Again, if you are "given that $ sumlimits_{n=1}^{N} a_n r_n=0$ for every natural $N$" then you know that $a_nr_n=0$ for every $n$. Thus, which parts of your question, if any, do not fall?
$endgroup$
– Did
Jan 2 at 22:36
$begingroup$
I think there is something missing in my work. I will be back after reviewing my note books.
$endgroup$
– ersh
Jan 2 at 22:38
$begingroup$
@Did I reviewed my work and I find this question is wrong. I took the index $N$ in a wrong direction in my work. Apparently my whole day wasted.
$endgroup$
– ersh
Jan 2 at 23:03
$begingroup$
I have edited my question. Does it make sense now? Now for fixed but arbitrary $N$ (and corresponding $a_i,r_i$), I am supposed to choose $t$ number of constants $b_i$ making that expression zero.
$endgroup$
– ersh
Jan 2 at 23:13
add a comment |
$begingroup$
I am trying to find the real constants $b_i$ such the $displaystyle sum_{n=1}^{N} a_n left(displaystyle sum_{n=1}^{t} b_i[r_n-(r_n)^2]+s(i))right)=0$ where I am given that $displaystyle sum_{n=1}^{N} a_n r_n=0$ for every natural $N$ and $s(i)$ is certain function of $i$ (I have the explicit form for $s(i)$ as well in terms of a series).
Note that $t$ is at our disposal. I tried to manipulate my expression for $t=3$ to find such constants $b_1,b_2,b_3$ unsuccessfully. I also want to mention that there is no guarantee for the existence of such constants except $b_i=0$ for all $i$.
EDIT: I reviewed my work and I found my original question was wrong. I took the index $N$ in a wrong direction. Now in above question $N$ (say $N=2$) is fixed so that $displaystyle sum_{n=1}^{2} a_n r_n=0$ with $a_1,a_2inmathbb{R}$ and $r_1,r_2in (0,1]$.
Now for fixed but arbitrary $N$ (and corresponding $a_i,r_i$), I am supposed to choose $t$ number of constants $b_i$ (not all zero) making that expression zero.
real-analysis sequences-and-series summation
asked Jan 2 at 22:32
ershersh
294112
$endgroup$
I am trying to find the real constants $b_i$ such the $displaystyle sum_{n=1}^{N} a_n left(displaystyle sum_{n=1}^{t} b_i[r_n-(r_n)^2]+s(i))right)=0$ where I am given that $displaystyle sum_{n=1}^{N} a_n r_n=0$ for every natural $N$ and $s(i)$ is certain function of $i$ (I have the explicit form for $s(i)$ as well in terms of a series).
Note that $t$ is at our disposal. I tried to manipulate my expression for $t=3$ to find such constants $b_1,b_2,b_3$ unsuccessfully. I also want to mention that there is no guarantee for the existence of such constants except $b_i=0$ for all $i$.
EDIT: I reviewed my work and I found my original question was wrong. I took the index $N$ in a wrong direction. Now in above question $N$ (say $N=2$) is fixed so that $displaystyle sum_{n=1}^{2} a_n r_n=0$ with $a_1,a_2inmathbb{R}$ and $r_1,r_2in (0,1]$.
Now for fixed but arbitrary $N$ (and corresponding $a_i,r_i$), I am supposed to choose $t$ number of constants $b_i$ (not all zero) making that expression zero.
real-analysis sequences-and-series summation
real-analysis sequences-and-series summation
asked Jan 2 at 22:32
ershersh
294112
asked Jan 2 at 22:32
ershersh
294112
edited Jan 2 at 23:18
ersh
asked Jan 2 at 22:32
ershersh
294112
asked Jan 2 at 22:32
ershersh
294112
asked Jan 2 at 22:32
ershersh
294112
294112
3
$begingroup$
Again, if you are "given that $ sumlimits_{n=1}^{N} a_n r_n=0$ for every natural $N$" then you know that $a_nr_n=0$ for every $n$. Thus, which parts of your question, if any, do not fall?
$endgroup$
– Did
Jan 2 at 22:36
$begingroup$
I think there is something missing in my work. I will be back after reviewing my note books.
$endgroup$
– ersh
Jan 2 at 22:38
$begingroup$
@Did I reviewed my work and I find this question is wrong. I took the index $N$ in a wrong direction in my work. Apparently my whole day wasted.
$endgroup$
– ersh
Jan 2 at 23:03
$begingroup$
I have edited my question. Does it make sense now? Now for fixed but arbitrary $N$ (and corresponding $a_i,r_i$), I am supposed to choose $t$ number of constants $b_i$ making that expression zero.
$endgroup$
– ersh
Jan 2 at 23:13
add a comment |
3
$begingroup$
Again, if you are "given that $ sumlimits_{n=1}^{N} a_n r_n=0$ for every natural $N$" then you know that $a_nr_n=0$ for every $n$. Thus, which parts of your question, if any, do not fall?
$endgroup$
– Did
Jan 2 at 22:36
$begingroup$
I think there is something missing in my work. I will be back after reviewing my note books.
$endgroup$
– ersh
Jan 2 at 22:38
$begingroup$
@Did I reviewed my work and I find this question is wrong. I took the index $N$ in a wrong direction in my work. Apparently my whole day wasted.
$endgroup$
– ersh
Jan 2 at 23:03
$begingroup$
I have edited my question. Does it make sense now? Now for fixed but arbitrary $N$ (and corresponding $a_i,r_i$), I am supposed to choose $t$ number of constants $b_i$ making that expression zero.
$endgroup$
– ersh
Jan 2 at 23:13
3
3
$begingroup$
Again, if you are "given that $ sumlimits_{n=1}^{N} a_n r_n=0$ for every natural $N$" then you know that $a_nr_n=0$ for every $n$. Thus, which parts of your question, if any, do not fall?
$endgroup$
– Did
Jan 2 at 22:36
$begingroup$
Again, if you are "given that $ sumlimits_{n=1}^{N} a_n r_n=0$ for every natural $N$" then you know that $a_nr_n=0$ for every $n$. Thus, which parts of your question, if any, do not fall?
$endgroup$
– Did
Jan 2 at 22:36
$begingroup$
I think there is something missing in my work. I will be back after reviewing my note books.
$endgroup$
– ersh
Jan 2 at 22:38
$begingroup$
I think there is something missing in my work. I will be back after reviewing my note books.
$endgroup$
– ersh
Jan 2 at 22:38
$begingroup$
@Did I reviewed my work and I find this question is wrong. I took the index $N$ in a wrong direction in my work. Apparently my whole day wasted.
$endgroup$
– ersh
Jan 2 at 23:03
$begingroup$
@Did I reviewed my work and I find this question is wrong. I took the index $N$ in a wrong direction in my work. Apparently my whole day wasted.
$endgroup$
– ersh
Jan 2 at 23:03
$begingroup$
I have edited my question. Does it make sense now? Now for fixed but arbitrary $N$ (and corresponding $a_i,r_i$), I am supposed to choose $t$ number of constants $b_i$ making that expression zero.
$endgroup$
– ersh
Jan 2 at 23:13
$begingroup$
I have edited my question. Does it make sense now? Now for fixed but arbitrary $N$ (and corresponding $a_i,r_i$), I am supposed to choose $t$ number of constants $b_i$ making that expression zero.
$endgroup$
– ersh
Jan 2 at 23:13
add a comment |
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$begingroup$
Again, if you are "given that $ sumlimits_{n=1}^{N} a_n r_n=0$ for every natural $N$" then you know that $a_nr_n=0$ for every $n$. Thus, which parts of your question, if any, do not fall?
$endgroup$
– Did
Jan 2 at 22:36
$begingroup$
I think there is something missing in my work. I will be back after reviewing my note books.
$endgroup$
– ersh
Jan 2 at 22:38
$begingroup$
@Did I reviewed my work and I find this question is wrong. I took the index $N$ in a wrong direction in my work. Apparently my whole day wasted.
$endgroup$
– ersh
Jan 2 at 23:03
$begingroup$
I have edited my question. Does it make sense now? Now for fixed but arbitrary $N$ (and corresponding $a_i,r_i$), I am supposed to choose $t$ number of constants $b_i$ making that expression zero.
$endgroup$
– ersh
Jan 2 at 23:13