General Formula for the Reciprocal of a Sum of Reciprocals

[FredericChopin]

I know that the reduced mass, μ, of an object is:

$$\mu = \frac{1}{\frac{1}{m_1} + \frac{1}{m_2}}$$

$$\mu = \frac{m_1 m_2}{ m_1 + m_2 }$$

But is there a general formula (or a simplified expression) for finding the value of:

$$\frac{1}{ \frac{1}{x_1} + \frac{1}{x_2} + ... + \frac{1}{x_n} }$$ ?

Thank you.

 

[matteo137]

This is the same as solving:
$$\dfrac{1}{x1}+\dfrac{1}{x2}+\dfrac{1}{x3}+... = \dfrac{A}{B}$$

$$\left( \dfrac{1}{x1}+\dfrac{1}{x2}+\dfrac{1}{x3}+... \right)^{-1}= \dfrac{\prod_1 x_i}{\sum_i \prod_{j\neq i} x_j}$$

 

[FredericChopin]

This is the same as solving:
$$\dfrac{1}{x1}+\dfrac{1}{x2}+\dfrac{1}{x3}+... = \dfrac{A}{B}$$

$$\left( \dfrac{1}{x1}+\dfrac{1}{x2}+\dfrac{1}{x3}+... \right)^{-1}= \dfrac{\prod_1 x_i}{\sum_i \prod_{j\neq i} x_j}$$

Great! Thank you so much!