General Formula for the Reciprocal of a Sum of Reciprocals

  • #1
I know that the reduced mass, μ, of an object is:

[tex]\mu = \frac{1}{\frac{1}{m_1} + \frac{1}{m_2}}[/tex]

[tex]\mu = \frac{m_1 m_2}{ m_1 + m_2 }[/tex]

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

[tex]\frac{1}{ \frac{1}{x_1} + \frac{1}{x_2} + ... + \frac{1}{x_n} }[/tex] ?

Thank you.
 
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Answers and Replies

  • #2
42
9
This is the same as solving:
[tex]\dfrac{1}{x1}+\dfrac{1}{x2}+\dfrac{1}{x3}+... = \dfrac{A}{B}[/tex]

[tex]\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}[/tex]
 
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  • #3
This is the same as solving:
[tex]\dfrac{1}{x1}+\dfrac{1}{x2}+\dfrac{1}{x3}+... = \dfrac{A}{B}[/tex]

[tex]\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}[/tex]
Great! Thank you so much!
 

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