Consider this sum of inverses:
$$\frac{1}{f_n} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \cdots + \frac{1}{R_{n}}$$
The sum can be "simplified" with summation symbol:
$$\frac{1}{f_n} = \sum_{i=1}^n\frac{1}{R_{i}}$$
I want to compute $f$ and express the above formula in terms of summation symbol(s).
Consider $n = 3$ case:
$$\begin{align*}f_n &= \frac{R_1R_2 + R_1R_3 + R_2R_3}{R_{1}R_{2}R_{3}} \\ &= \frac{\color{red}{?}}{\prod_{i=1}^n R_i}\end{align*}$$
I am not sure how to express the $\color{red}{?}$ numerator in terms of, probably, summation symbol(s).
There is a pattern for sure.
Consider $n = 4$:
$$f_n = \frac{\overbrace{\overbrace{R_{1}R_{2}R_{3}}^{n-1 \text{ terms}} + \overbrace{R_{1}R_{2}R_{4}}^{n-1 \text{ terms}} + \overbrace{R_{1}R_{3}R_{4}}^{n-1 \text{ terms}} + \overbrace{R_{2}R_{3}R_{4}}^{n-1 \text{ terms}}}^{n \text{ terms}}}{\prod_{i=1}^{n} R_{i}}$$
Would anyone be able to help me formalize this numerator for $n \in \mathbb{Z}^{+}$?