Convergence of Sum of Squares of Sequence with Fixed Sum
$begingroup$
For any given $n$, suppose we have a partition of $1$, i.e. a finite sequence $a_{n,1}, a_{n,2}, ldots, a_{n,n}$ such that $0 leq a_{n,i} < 1$ for all $i$ and $sum_{i=1}^n a_{n,i} = 1$. Suppose also that this partition gets increasingly fine: formally, $max_{1leq ileq n} a_{n,i} to 0$ as $ntoinfty$. Is it true that $sum_{i=1}^n a_{n,i}^2 to 0$ as $ntoinfty$? If so, how can I prove this?
sequences-and-series
asked Nov 29 '18 at 0:05
EmpiromancerEmpiromancer
663315
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add a comment |
$begingroup$
For any given $n$, suppose we have a partition of $1$, i.e. a finite sequence $a_{n,1}, a_{n,2}, ldots, a_{n,n}$ such that $0 leq a_{n,i} < 1$ for all $i$ and $sum_{i=1}^n a_{n,i} = 1$. Suppose also that this partition gets increasingly fine: formally, $max_{1leq ileq n} a_{n,i} to 0$ as $ntoinfty$. Is it true that $sum_{i=1}^n a_{n,i}^2 to 0$ as $ntoinfty$? If so, how can I prove this?
sequences-and-series
asked Nov 29 '18 at 0:05
EmpiromancerEmpiromancer
663315
$endgroup$
add a comment |
$begingroup$
For any given $n$, suppose we have a partition of $1$, i.e. a finite sequence $a_{n,1}, a_{n,2}, ldots, a_{n,n}$ such that $0 leq a_{n,i} < 1$ for all $i$ and $sum_{i=1}^n a_{n,i} = 1$. Suppose also that this partition gets increasingly fine: formally, $max_{1leq ileq n} a_{n,i} to 0$ as $ntoinfty$. Is it true that $sum_{i=1}^n a_{n,i}^2 to 0$ as $ntoinfty$? If so, how can I prove this?
sequences-and-series
asked Nov 29 '18 at 0:05
EmpiromancerEmpiromancer
663315
$endgroup$
For any given $n$, suppose we have a partition of $1$, i.e. a finite sequence $a_{n,1}, a_{n,2}, ldots, a_{n,n}$ such that $0 leq a_{n,i} < 1$ for all $i$ and $sum_{i=1}^n a_{n,i} = 1$. Suppose also that this partition gets increasingly fine: formally, $max_{1leq ileq n} a_{n,i} to 0$ as $ntoinfty$. Is it true that $sum_{i=1}^n a_{n,i}^2 to 0$ as $ntoinfty$? If so, how can I prove this?
sequences-and-series
sequences-and-series
asked Nov 29 '18 at 0:05
EmpiromancerEmpiromancer
663315
asked Nov 29 '18 at 0:05
EmpiromancerEmpiromancer
663315
asked Nov 29 '18 at 0:05
EmpiromancerEmpiromancer
663315
asked Nov 29 '18 at 0:05
EmpiromancerEmpiromancer
663315
asked Nov 29 '18 at 0:05
EmpiromancerEmpiromancer
663315
663315
add a comment |
add a comment |
1 Answer
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$sum_{i=1}^{n} a_{n,i}^{2} leq (max_{1leq i leq n} a_{n,i}) sum_{i=1}^{n} a_{n,i}=max_{1leq i leq n} a_{n,i} to 0$.
answered Nov 29 '18 at 0:17
Kavi Rama MurthyKavi Rama Murthy
52.9k32055
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1 Answer
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1 Answer
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$begingroup$
$sum_{i=1}^{n} a_{n,i}^{2} leq (max_{1leq i leq n} a_{n,i}) sum_{i=1}^{n} a_{n,i}=max_{1leq i leq n} a_{n,i} to 0$.
answered Nov 29 '18 at 0:17
Kavi Rama MurthyKavi Rama Murthy
52.9k32055
$endgroup$
add a comment |
$begingroup$
$sum_{i=1}^{n} a_{n,i}^{2} leq (max_{1leq i leq n} a_{n,i}) sum_{i=1}^{n} a_{n,i}=max_{1leq i leq n} a_{n,i} to 0$.
answered Nov 29 '18 at 0:17
Kavi Rama MurthyKavi Rama Murthy
52.9k32055
$endgroup$
add a comment |
$begingroup$
$sum_{i=1}^{n} a_{n,i}^{2} leq (max_{1leq i leq n} a_{n,i}) sum_{i=1}^{n} a_{n,i}=max_{1leq i leq n} a_{n,i} to 0$.
answered Nov 29 '18 at 0:17
Kavi Rama MurthyKavi Rama Murthy
52.9k32055
$endgroup$
$sum_{i=1}^{n} a_{n,i}^{2} leq (max_{1leq i leq n} a_{n,i}) sum_{i=1}^{n} a_{n,i}=max_{1leq i leq n} a_{n,i} to 0$.
answered Nov 29 '18 at 0:17
Kavi Rama MurthyKavi Rama Murthy
52.9k32055
answered Nov 29 '18 at 0:17
Kavi Rama MurthyKavi Rama Murthy
52.9k32055
answered Nov 29 '18 at 0:17
Kavi Rama MurthyKavi Rama Murthy
52.9k32055
answered Nov 29 '18 at 0:17
Kavi Rama MurthyKavi Rama Murthy
52.9k32055
52.9k32055
add a comment |
add a comment |
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