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Hello, Suppose I have two sequences $a_n, b_n$ with $n\ge 1$ and $a_n, b_n \ge0$. I know that $\sum_{n=1}^{k} a_n \le \sum_{n=1}^{k} b_n$ for any $k > 0$ and $\sum _ {n=1}^{\infty} b_n$ exists and is finite.

Can I obtain an upper bound on $a_n$, like $a_n\le C b_n \quad \forall n$?

Thank you

Best,

Michele

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 Without motivation, this question doesn't belong here. – Anthony Quas Jan 21 at 20:11
No. Take any integer $m>C$, with sequences $a_n=0$ for $n\neq m$, $a_m=m$; $b_n=1$ if $n \leq m$, and $0$ otherwise. – Zack Wolske Jan 21 at 21:38
or $a_n = \frac{1}{n^2}$ and $b_1 = \frac{\pi^2}{6}$, $b_n = \frac{1}{n^3}$ $(n>1)$ – njguliyev Jan 21 at 21:43
In any case, you can't. Take e.g. $b_n=1/n^2 $ and $a_n=1/n$ if $n$ is a perfect square, , $0$ otherwise. – Pietro Majer Jan 21 at 21:46

closed as off topic by Anthony Quas, Goldstern, Andreas Blass, Bill Johnson, Pietro Majer Jan 21 at 21:47

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