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$












1












$begingroup$


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







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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 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
















1












$begingroup$


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







share|cite|improve this question











$endgroup$








  • 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 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














1












1








1





$begingroup$


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







share|cite|improve this question











$endgroup$




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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 2 at 22:36







ersh

















asked Jan 2 at 21:41









ershersh

423113




423113








  • 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 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














  • 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 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








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










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