If $ sumlimits_{n=1}^{N} a_n r_n=0$, what can we say about $ sumlimits_{n=1}^{N} a_n (r_n)^2=0$
$begingroup$
If $displaystyle sum_{n=1}^{N} a_n r_n=0$ for every natural $N$ where $a_nin mathbb{R}$ and $r_nin(0,1]$, can we conclude anything significant about $displaystyle sum_{n=1}^{N} a_n (r_n)^2,displaystyle sum_{n=1}^{N} a_n (r_n)^3.. $?
EDIT: The related and completed question is in my new post here: (Find constants $b_i$ such that $sum_{n=1}^{N} a_n (sum_{n=1}^{t} b_i[r_n-(r_n)^2]+s(i))=0$)
real-analysis summation
asked Jan 2 at 21:41
ershersh
438113
$endgroup$
add a comment |
$begingroup$
If $displaystyle sum_{n=1}^{N} a_n r_n=0$ for every natural $N$ where $a_nin mathbb{R}$ and $r_nin(0,1]$, can we conclude anything significant about $displaystyle sum_{n=1}^{N} a_n (r_n)^2,displaystyle sum_{n=1}^{N} a_n (r_n)^3.. $?
EDIT: The related and completed question is in my new post here: (Find constants $b_i$ such that $sum_{n=1}^{N} a_n (sum_{n=1}^{t} b_i[r_n-(r_n)^2]+s(i))=0$)
real-analysis summation
asked Jan 2 at 21:41
ershersh
438113
$endgroup$
11
$begingroup$
If $sum_{n=1}^{N}{a_nr_n}=0$ for every $N$, doesn't that mean $a_Nr_N=0$ for every $N$?
$endgroup$
– Ant
Jan 2 at 21:46
1
$begingroup$
Yessssssssssss!
$endgroup$
– ersh
Jan 2 at 22:09
1
$begingroup$
Thus, no question?
$endgroup$
– Did
Jan 2 at 22:31
$begingroup$
Please do not usedisplaystylein titles.
$endgroup$
– Did
Jan 2 at 22:31
$begingroup$
@Winther Thanks, I will be reviving my work and notebook first. I think there is something missing.
$endgroup$
– ersh
Jan 2 at 22:40
add a comment |
$begingroup$
If $displaystyle sum_{n=1}^{N} a_n r_n=0$ for every natural $N$ where $a_nin mathbb{R}$ and $r_nin(0,1]$, can we conclude anything significant about $displaystyle sum_{n=1}^{N} a_n (r_n)^2,displaystyle sum_{n=1}^{N} a_n (r_n)^3.. $?
EDIT: The related and completed question is in my new post here: (Find constants $b_i$ such that $sum_{n=1}^{N} a_n (sum_{n=1}^{t} b_i[r_n-(r_n)^2]+s(i))=0$)
real-analysis summation
asked Jan 2 at 21:41
ershersh
438113
$endgroup$
If $displaystyle sum_{n=1}^{N} a_n r_n=0$ for every natural $N$ where $a_nin mathbb{R}$ and $r_nin(0,1]$, can we conclude anything significant about $displaystyle sum_{n=1}^{N} a_n (r_n)^2,displaystyle sum_{n=1}^{N} a_n (r_n)^3.. $?
EDIT: The related and completed question is in my new post here: (Find constants $b_i$ such that $sum_{n=1}^{N} a_n (sum_{n=1}^{t} b_i[r_n-(r_n)^2]+s(i))=0$)
real-analysis summation
real-analysis summation
asked Jan 2 at 21:41
ershersh
438113
asked Jan 2 at 21:41
ershersh
438113
edited Jan 2 at 22:36
ersh
asked Jan 2 at 21:41
ershersh
438113
asked Jan 2 at 21:41
ershersh
438113
asked Jan 2 at 21:41
ershersh
438113
438113
11
$begingroup$
If $sum_{n=1}^{N}{a_nr_n}=0$ for every $N$, doesn't that mean $a_Nr_N=0$ for every $N$?
$endgroup$
– Ant
Jan 2 at 21:46
1
$begingroup$
Yessssssssssss!
$endgroup$
– ersh
Jan 2 at 22:09
1
$begingroup$
Thus, no question?
$endgroup$
– Did
Jan 2 at 22:31
$begingroup$
Please do not usedisplaystylein titles.
$endgroup$
– Did
Jan 2 at 22:31
$begingroup$
@Winther Thanks, I will be reviving my work and notebook first. I think there is something missing.
$endgroup$
– ersh
Jan 2 at 22:40
add a comment |
11
$begingroup$
If $sum_{n=1}^{N}{a_nr_n}=0$ for every $N$, doesn't that mean $a_Nr_N=0$ for every $N$?
$endgroup$
– Ant
Jan 2 at 21:46
1
$begingroup$
Yessssssssssss!
$endgroup$
– ersh
Jan 2 at 22:09
1
$begingroup$
Thus, no question?
$endgroup$
– Did
Jan 2 at 22:31
$begingroup$
Please do not usedisplaystylein titles.
$endgroup$
– Did
Jan 2 at 22:31
$begingroup$
@Winther Thanks, I will be reviving my work and notebook first. I think there is something missing.
$endgroup$
– ersh
Jan 2 at 22:40
11
11
$begingroup$
If $sum_{n=1}^{N}{a_nr_n}=0$ for every $N$, doesn't that mean $a_Nr_N=0$ for every $N$?
$endgroup$
– Ant
Jan 2 at 21:46
$begingroup$
If $sum_{n=1}^{N}{a_nr_n}=0$ for every $N$, doesn't that mean $a_Nr_N=0$ for every $N$?
$endgroup$
– Ant
Jan 2 at 21:46
1
1
$begingroup$
Yessssssssssss!
$endgroup$
– ersh
Jan 2 at 22:09
$begingroup$
Yessssssssssss!
$endgroup$
– ersh
Jan 2 at 22:09
1
1
$begingroup$
Thus, no question?
$endgroup$
– Did
Jan 2 at 22:31
$begingroup$
Thus, no question?
$endgroup$
– Did
Jan 2 at 22:31
$begingroup$
Please do not use
displaystyle in titles.$endgroup$
– Did
Jan 2 at 22:31
$begingroup$
Please do not use
displaystyle in titles.$endgroup$
– Did
Jan 2 at 22:31
$begingroup$
@Winther Thanks, I will be reviving my work and notebook first. I think there is something missing.
$endgroup$
– ersh
Jan 2 at 22:40
$begingroup$
@Winther Thanks, I will be reviving my work and notebook first. I think there is something missing.
$endgroup$
– ersh
Jan 2 at 22:40
add a comment |
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11
$begingroup$
If $sum_{n=1}^{N}{a_nr_n}=0$ for every $N$, doesn't that mean $a_Nr_N=0$ for every $N$?
$endgroup$
– Ant
Jan 2 at 21:46
1
$begingroup$
Yessssssssssss!
$endgroup$
– ersh
Jan 2 at 22:09
1
$begingroup$
Thus, no question?
$endgroup$
– Did
Jan 2 at 22:31
$begingroup$
Please do not use
displaystylein titles.$endgroup$
– Did
Jan 2 at 22:31
$begingroup$
@Winther Thanks, I will be reviving my work and notebook first. I think there is something missing.
$endgroup$
– ersh
Jan 2 at 22:40