Let $a_n$ be a sequence of positive numbers such that the series $\sum \limits_{n=1}^{\infty} a_n$ converges. Prove that there does not exist a positive real number $\ell$ such that:
$$\sum_{n=1}^{\infty}a_n \leq \ell \sum_{n=1}^{\infty}\frac{a_n}{\sum \limits_{k=1}^{n}\frac{1}{a_k}} $$
Solution
Suppose $a_n$ is a sequence such that $\sum \limits_{n=1}^{\infty}a_n=A$ and $\sum \limits_{n=1}^{\infty}\frac{a_n}{\sum \limits_{k=1}^n1/a_k}=B.$ Note that $A$ and $B,$ which are assumed to be finite, must be positive. Now let $\lambda$ be an arbitrary positive constant. Then $\sum \limits_{n=1}^{\infty}\lambda a_n=\lambda A$ and $\sum \limits_{n=1}^{\infty}\frac{\lambda a_n}{\sum \limits_{k=1}^n1/(\lambda a_k)}=\lambda^2B.$
If the proposed inequality were true, we would have to have $\lambda A\le \ell \lambda^2B$ for all $\lambda>0.$ Now let $\lambda\to 0^+$ and note that there is no fixed $\ell$ for which this inequality can hold.
$$\sum_{n=1}^{\infty}a_n \leq \ell \sum_{n=1}^{\infty}\frac{a_n}{\sum \limits_{k=1}^{n}\frac{1}{a_k}} $$
Solution
Suppose $a_n$ is a sequence such that $\sum \limits_{n=1}^{\infty}a_n=A$ and $\sum \limits_{n=1}^{\infty}\frac{a_n}{\sum \limits_{k=1}^n1/a_k}=B.$ Note that $A$ and $B,$ which are assumed to be finite, must be positive. Now let $\lambda$ be an arbitrary positive constant. Then $\sum \limits_{n=1}^{\infty}\lambda a_n=\lambda A$ and $\sum \limits_{n=1}^{\infty}\frac{\lambda a_n}{\sum \limits_{k=1}^n1/(\lambda a_k)}=\lambda^2B.$
If the proposed inequality were true, we would have to have $\lambda A\le \ell \lambda^2B$ for all $\lambda>0.$ Now let $\lambda\to 0^+$ and note that there is no fixed $\ell$ for which this inequality can hold.
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