I am currently taking a class that involves studying circuits. We recently went over the equation for resistance in a circuit with parallel resistors, where $n$ is the total amount of resistors in parallel $$ \frac{1}{R_{Total}} = \frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}}+\cdots+\frac{1}{R_{n}}. $$ This is the equation I've derived from the parallel resistance formula. $R _i$ represents the resistance of resistor $i$, and $n$ represents the total amount of resistors in parallel. It solves for the total resistance. $$ R_{Total}=\frac{\prod_{i=1}^{n}R_i}{\sum_{i=1}^{n}R_i\sum_{v=i+1}^{n}R_v} $$
From this formula, taking it out of context, I have two questions:
- Does this formula actually hold true?
- If it does, suppose there were an infinite amount of resistors in parallel, where $n \to \infty$. Is it possible to check for the convergence of this whole statement?