Perhaps this is a redundant question, but regardless:

Say that I have been given the function: $$f(x) = -ln(2) + \sum_{n=1}^∞ \frac{x^n}{2^n n}$$

By the ratio test, the power series part converges for $|x| < 2$.

I have to show that $f'(x) = \frac{1}{2-x}$

The first and most obvious thing to do is differentiate $-ln(2)$, which simply is $0$ (it's a constant). But how would you differentiate a power series which you have not been given the initial representation of? Would I have to find the value of the power series, and differentiate that, or integrate $f'(x)$, assume that as the representation of $\sum_{n=1}^∞ \frac{x^n}{2^n n}$ and go from there? The latter just seems like a bit of a circular proof.

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    $\begingroup$ Inside its radius of convergence, one can differentiate power series term wise. $\endgroup$ – Lutz Lehmann Sep 26 '16 at 11:47


When $|x|<2$

  • $\begingroup$ Yeah, this question was pretty redundant... $\endgroup$ – R. Rengold Sep 26 '16 at 14:10

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