TITLE: How would you solve an equation of the form $e^{-x} - \sin(x) = 0$?
QUESTION [6 upvotes]: I've been trying to do this for ages. I'm worried that it's impossible, but I have heard that it can be done by hand. 
As long as I can get $x$ by hand I can obviously work out its value via calculator.
Please help, I am so stuck and desperately need assistance.
Thanks in advance!

REPLY [2 votes]: If you build the series expansion of the rhs of
$$y=e^{-x}-\sin(x)$$ around $x=n \pi$ and later series reversion forcing $y=0$, you obtain
$$x_n=n\pi+\frac{1}{1+e^{(1+i) \pi  n}}+\frac{1}{2 \left(1+e^{(1+i) \pi  n}\right)^3}+\cdots$$
$$\left(
\begin{array}{ccc}
 n & \text{approximation} & \text{solution} \\
 0 & 0.5625000000000000000 &  0.5885327439818610774 \\
 1 & 3.0963808805403588820 &  3.0963639324106461156 \\
 2 & 6.2850492723072431593 &  6.2850492733825865338 \\
 3 & 9.4246972547386088459 &  9.4246972547385212191 \\
 4 & 12.566374101689367670 &  12.566374101689367677 \\
 5 & 15.707963117247215942 &  15.707963117247215942 \\
 6 & 18.849555928051171524 &  18.849555928051171524 \\
 7 & 21.991148574847125823 &  21.991148574847125823 \\
 8 & 25.132741228730507464 &  25.132741228730507464 \\
 9 & 28.274333882307613598 &  28.274333882307613598 \\
 10 & 31.415926535897955096 &  31.415926535897955096
\end{array}
\right)$$