What does this infinite sum represent?

Is there a better way to write the following?

$$\sum_{i=0}^\infty \left(\frac{2}{3}\right)^i \frac{\left(\dot{f}\right)^i}{f^{2i-1}},$$

where $\dot{f} = df/dt$.

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2 Answers

It's $$\dfrac{f^3}{f^2 - (2/3) \dot{f}}$$ whenever the infinite series converges.

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Hint: you can write $$\left( \dfrac{2}{3} \right)^i \dfrac{(\dot f)^i}{f^{2i-1}} = \left( \dfrac{2}{3} \dfrac{\dot f}{f^2} \right)^i \cdot f$$ so, presuming the sum converges, what you have is a geometric series.

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