# Evaluate $\sum\limits_{r=1}^\infty(-1)^{r+1}\frac{\cos(2r-1)x}{2r-1}$

I would like to know how to evaluate $$\sum\limits_{r=1}^\infty(-1)^{r+1}\frac{\cos(2r-1)x}{2r-1}$$

There are a couple of issues I have with this. Firstly, depending on the value of $$x$$, it seems, at least numerically, that this value is always $$\pm\sqrt{\frac{\sqrt5-1}{2}}$$ If this is true, how would I prove this?

Secondly, I tried using complex numbers to evalaute this, as shown briefly below:

$$C=\sum\limits_{r=1}^\infty(-1)^{r+1}\frac{\cos(2r-1)x}{2r-1}$$ $$S=\sum\limits_{r=1}^\infty(-1)^{r+1}\frac{\sin(2r-1)x}{2r-1}$$ $$\implies C+iS=\arctan e^{ix}$$ on using the power series of $$\arctan x$$. Differentiating yields a completely imaginary number: $$\frac{i}{2}\sec x$$ which proves that the value of $$C$$ is a constant. However, this gives me no information whatsoever on the value of $$C$$.

Thank you for your help.

• Why would you want the exact value of this series? Or are you only interested in some "properties" of this series?
– user9464
Jan 16 at 23:40
• @mrsamy I'm just looking at it out of curiosity. Jan 16 at 23:41
• That's a fair reason :-)
– user9464
Jan 16 at 23:41
• Have you studied some Fourier Series, perhaps in advanced analysis...or harmonic analysis...or stuff? Jan 17 at 0:41
• @DonAntonio unfortunately not; I am still in high school and don't know a lot beyond the high school curriculum. Jan 17 at 9:35

As you noted, the real part of the sum, $$C$$, is constant no matter what the value of $$x$$ (so long as $$x$$ is real). Thus plug in $$x = 0$$

$$C = \operatorname{Re}(\arctan e^{i0}) = \frac{\pi}{4}$$

• Thanks so much! However, it seems that $C$ can take different values; namely $$\pm\frac{\pi}{4}$$ at least according to desmos. How do I obtain the negative value? Jan 16 at 23:48
• ... no matter what the value of $x$... - not quite (replace $x$ by $x+\pi$ in the original series to see what actually happens). And... actually we reinvent this... or more specifically this. Jan 16 at 23:50
• @metamorphy I see, thanks for the clarification! Jan 16 at 23:52
• @A-LevelStudent the sum changes values at $x=\frac{\pi}{2}$ because arctan is undefined at $i$. It's like what happens with the graph of $\arctan(2\tan x)$ at the same points. But thanks for pointing it out, I would not have caught that otherwise. Jan 16 at 23:55
• Why isn't $\arctan$ defined at $i$? Isn't it just equal to $i~\text{artanh}1$? Jan 16 at 23:56

The Series

This series only converges for $$x\in\mathbb{R}$$. If $$x\not\in\mathbb{R}$$, the terms do not go to $$0$$.$$\newcommand{\sgn}{\operatorname{sgn}}\newcommand{\Re}{\operatorname{Re}}\newcommand{\Im}{\operatorname{Im}}$$ \begin{align} &\sum_{r=1}^\infty(-1)^{r+1}\frac{\cos((2r-1)x)}{2r-1}\tag{1a}\\ &=\frac12\sum_{r=1}^\infty(-1)^{r+1}\frac{e^{i(2r-1)x}}{2r-1}+\frac12\sum_{r=1}^\infty(-1)^{r+1}\frac{e^{-i(2r-1)x}}{2r-1}\tag{1b}\\ &=\frac12\tan^{-1}\left(e^{ix}\right)+\frac12\tan^{-1}\left(e^{-ix}\right)\tag{1c}\\[9pt] &=\Re\left(\tan^{-1}\left(e^{ix}\right)\right)\tag{1d}\\[12pt] &=\frac\pi4\,\sgn(\cos(x))\tag{1e} \end{align} Explanation:
$$\text{(1b)}$$: $$\cos(x)=\frac12\left(e^{ix}+e^{-ix}\right)$$
$$\text{(1c)}$$: power series for $$\tan^{-1}(z)$$
$$\text{(1d)}$$: $$\tan(\bar z)=\overline{\tan(z)}$$
$$\text{(1e)}$$: $$|\tan(z)\,|=1$$ when $$\Re(z)=\pm\frac\pi4$$
$$\phantom{\text{(1e):}}$$ and $$\Re(\tan(z))$$ has the same sign as $$\Re(z)$$
$$\phantom{\text{(1e):}}$$ also, if $$\cos(x)=0$$, all the terms of the series are $$0$$

Concerning $$\boldsymbol{\tan(x+iy)}$$

Note that \begin{align} \tan(x+iy)&=\frac{\sin(x)\cosh(y)+i\cos(x)\sinh(y)}{\cos(x)\cosh(y)-i\sin(x)\sinh(y)}\tag{2a}\\ &=\frac{\sin(x)\cos(x)+i\sinh(y)\cosh(y)}{\cos^2(x)\cosh^2(y)+\sin^2(x)\sinh^2(y)}\tag{2b} \end{align} is on the unit circle when the absolute values of the numerator and denominator are equal. Since $$\cosh^2(y)-\sinh^2(y)=1$$, equation $$\text{(2a)}$$ says that is when \begin{align} \sin^2(x)\cosh^2(y)+\cos^2(x)\sinh^2(y)&=\cos^2(x)\cosh^2(y)+\sin^2(x)\sinh^2(y)\tag{3a}\\[6pt] \sin^2(x)&=\cos^2(x)\tag{3b} \end{align} Thus, $$|\tan(z)\,|=1$$ when $$\Re(z)=\pm\frac\pi4$$

Furthermore, equation $$\text{(2b)}$$ says not only that $$\tan(\bar z)=\overline{\tan(z)}$$, but also that, when $$\Re(z)=\pm\frac\pi4$$, $$\Re(\tan(z))$$ has the same sign as $$\Re(z)$$.

• This answer has finally explained to me why $\tan (\bar {z})=\overline{\tan(z)}$. Thank you so much! Mar 23 at 22:55
• Actually $f(\bar z)=\overline{f(z)}$ is true for any function that is analytic on a connected region that includes a real interval and is real-valued on that real interval. See property of a holomorphic function on a connected open subset of complex plane.
– robjohn
Mar 24 at 7:55

I've learned some new related facts since posting the question, one of which solves this very simply. The relevant formula, stated below, can be found by using the fact $$\arctan z=-i\operatorname{artanh} iz$$. $$\arctan(x+iy)=\frac{1}{2}\arctan\frac{2x}{1-x^2-y^2}+\frac{i}{2}\operatorname{artanh}\frac{2y}{1+x^2+y^2}$$ Plugging in $$x=\cos x$$ and $$y=\sin x$$ we find that $$\arctan e^{ix}=\frac{1}{2}\arctan(\pm\infty)+\frac{i}{2}\operatorname{artanh}\sin x$$ And thus $$C=\pm\frac{\pi}{4}$$