TITLE: Is there any known relationship between $\sum_{i=1}^{n} f(i)$ and $\sum_{i=1}^{n} \dfrac {1}{f(i)}$
QUESTION [6 upvotes]: If we knew $\sum_{i=1}^{n} f(i)=S$  ($n$ can be $\infty$), is there anything we could say about $\sum_{i=1}^{n} \dfrac {1}{f(i)}$ in terms of $S$?  The only thing I was able to find on the web is link to a dead question. If there is no known relationship, is it possible for there to exist one that we just haven't discovered?

REPLY [7 votes]: $\newcommand{\angles}[1]{\left\langle\,{#1}\,\right\rangle}
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\begin{align}
\color{#f00}{\sum_{1 = 1}^{n}\mathrm{f}\pars{i} + \sum_{1 = 1}^{n}{1 \over \mathrm{f}\pars{i}}} & \geq \color{#f00}{2n}
\end{align}