Chapter 13.2, Problem 6P

Show that the series in (2.12) can be summed to get $T=\frac{200}{\pi }\arctan \left(\frac{\sin (\pi x/10)}{\sinh (\pi y/10)}\right)$

(with the are tangent in radians). Use this formula to check the value $T={26.1}^{°}$ at $x=y=5.$ Hints for summing the series: Use $\sin (n\pi x/10)=\mathrm{Im}{e}^{in\pi x/10}$ to write the series as Im ${\displaystyle \underset{\text{odd }n}{\sum }{z}^n}/n.$ (What is $z$ ?) Compare this with the series for $\ln [(1+z)/(1-z)]$ (see Chapter 1, Problem 13.17 ). Then use (13.5) of Chapter 2.