Maclaurin series similar to those of e^x

Chruser · 01:35 AM

$\reverse f_{g(n)}(x) = -\sum_{n=1}^{\infty} \frac{x^{g(n)}}{(g(n))}$ has some interesting ones.

$\reverse f_n(x) = \ln \left(1 - x \right)$

$\reverse f_{2n}(x) = \frac{1}{2} \ln \left( 1-x^2 \right)$

$\reverse f_{2n+1}(x) = x- \arctan (x)$

For $\reverse f_{2n+k}(x)$ where k is odd an arctan term shows up, sometimes added to a polynomial and/or logarithmic terms. For even k I see only polynomial and logarithmic terms. k = 7 has a much neater form than the others that I've looked at

$\reverse f_{2n+7}(x) = \frac{x^7}{7} + \frac{x^5}{5} + \frac{x^3}{3} + x - \arctan (x)$

$\reverse f_{kn}(x) = \frac{1}{k} \ln \left( 1-x^k \right)$

There may be some other stuff that exists.

Curious that the "base" series is that for the natural log, but an arctan shows up when modifying that series. I remember seeing another relation between ln and arctan as well:

$\reverse \arctan(x)=\frac{1}{2i}\ln \left(\frac{x-i}{x+i} \right)\ + \frac{\pi}{2}$ (for x>0...switch the sign on the pi/2 for negative x.)

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