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Let $f(x)$ be $\sum_{k=1}^{\infty} x^{2k+1}$. This sum equals $$x + x^3 + x^5 + \dots - x= x(1 + x^2 + (x^2)^2 + \dots) - x = \frac{x}{1-x^2} - x = \frac{x^3}{1-x^2}$$

Now, okay, the original sum is the derivative of $f$, but I don't think it is the conclusion the author expected...?

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  • $\begingroup$ I think that is the expected conclusion. $\endgroup$ Feb 3 '14 at 21:36
  • $\begingroup$ Actualy, I think you are expected to compute that derivative. $\endgroup$ Feb 3 '14 at 21:36
  • $\begingroup$ Okay, since this exercise (from "Introduction to algorithms" (Cormen et al.)) is star-marked, I expected something more difficult. Thank you for your answers. $\endgroup$
    – Greg82
    Feb 3 '14 at 21:47
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Yes you are allowed to differentiate, as $f(x)$ is differentiable for every $|x|<1$.

Then $$ f'(x)=\frac{3x^2}{1-x^2}+\frac{2x^4}{(1-x^2)^2}. $$

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