2016.3.4 Question 4

1. Notice that

   $$\frac{1}{1+x^r}-\frac{1}{1+x^{r+1}}=\frac{x^{r+1}-x^r}{(1+x^r)(1+x^{r+1})}=\frac{x^r(x-1)}{(1+x^r)(1+x^{r+1})}.$$

   Therefore, we have

   $$\begin{array}{llll}\hfill \sum _{r=1}^N\frac{x^r}{(1+x^r)(1+x^{r+1})}& =\sum _{r=1}^N\frac{1}{x-1}\left[\frac{1}{1+x^r}-\frac{1}{1+x^{r+1}}\right]& \hfill & \\ \hfill & =\frac{1}{x-1}\sum _{r=1}^N\left[\frac{1}{1+x^r}-\frac{1}{1+x^{r+1}}\right]& \hfill & \\ \hfill & =\frac{1}{x-1}\left[\frac{1}{1+x}-\frac{1}{1+x^{n+1}}\right].& \hfill & \end{array}$$

   For $\left|x\right|<1$, as $n\to \infty$, $x^{n+1}\to 0$. Therefore,

   $$\begin{array}{llll}\hfill \sum _{r=1}^{\infty }\frac{x^r}{(1+x^r)(1+x^{r+1})}& =\frac{1}{x-1}\left[\frac{1}{1+x}-1\right]& \hfill & \\ \hfill & =\frac{1}{x-1}\cdot \frac{-x}{1+x}& \hfill & \\ \hfill & =\frac{x}{1-x^2}& \hfill & \end{array}$$

   as desired.

2. Notice that

   $$\begin{array}{llll}\hfill \text{sech}<!--\; FUNCTION\; APPLICATION\; -->(\mathit{ry})\text{sech}<!--\; FUNCTION\; APPLICATION\; -->((r+1)y)& =\frac{2}{e^{\mathit{ry}}+e^{-\mathit{ry}}}\cdot \frac{2}{e^{(r+1)y}+e^{-(r+1)y}}& \hfill & \\ \hfill & =\frac{4e^{-\mathit{ry}-(r+1)y}}{\left(1+e^{-2\mathit{ry}}\right)\left(1+e^{-2(r+1)y}\right)}& \hfill & \\ \hfill & =4e^{-y}\frac{e^{-2\mathit{ry}}}{\left(1+e^{-2\mathit{ry}}\right)\left(1+e^{-2(r+1)y}\right)}.& \hfill & \end{array}$$

   Let $x=e^{-2y}$. We have

   $$\text{sech}<!--\; FUNCTION\; APPLICATION\; -->(\mathit{ry})\text{sech}<!--\; FUNCTION\; APPLICATION\; -->((r+1)y)=4e^{-y}\frac{x^r}{\left(1+x^r\right)\left(1+x^{r+1}\right)}.$$

   When $y>0$, $x=e^{-2y}\in (0,1)$. Therefore,

   $$\begin{array}{llll}\hfill \sum _{r=1}^{\infty }\text{sech}<!--\; FUNCTION\; APPLICATION\; -->(\mathit{ry})\text{sech}<!--\; FUNCTION\; APPLICATION\; -->((r+1)y)& =4e^{-y}\frac{e^{-2y}}{1-e^{-4y}}& \hfill & \\ \hfill & =2e^{-y}\frac{2}{e^{2y}-e^{-2y}}& \hfill & \\ \hfill & =2e^{-y}\text{cosech}<!--\; FUNCTION\; APPLICATION\; -->(2y)& \hfill & \end{array}$$

   as desired.

   Notice that for all $x\in ℝ$, $\text{cosh}<!--\; FUNCTION\; APPLICATION\; -->x=\text{cosh}<!--\; FUNCTION\; APPLICATION\; -->(-x)$, therefore $\text{sech}<!--\; FUNCTION\; APPLICATION\; -->x=\text{sech}<!--\; FUNCTION\; APPLICATION\; -->(-x)$.

   Therefore,

   $$\begin{array}{llll}\hfill & \sum _{r=-\infty }^{\infty }\text{sech}<!--\; FUNCTION\; APPLICATION\; -->(\mathit{ry})\text{sech}<!--\; FUNCTION\; APPLICATION\; -->((r+1)y)& \hfill & \\ \hfill & =\sum _{r=1}^{\infty }\text{sech}<!--\; FUNCTION\; APPLICATION\; -->(\mathit{ry})\text{sech}<!--\; FUNCTION\; APPLICATION\; -->((r+1)y)+\sum _{r=-\infty }^0\text{sech}<!--\; FUNCTION\; APPLICATION\; -->(\mathit{ry})\text{sech}<!--\; FUNCTION\; APPLICATION\; -->((r+1)y)& \hfill & \\ \hfill & =\sum _{r=1}^{\infty }\text{sech}<!--\; FUNCTION\; APPLICATION\; -->(\mathit{ry})\text{sech}<!--\; FUNCTION\; APPLICATION\; -->((r+1)y)+\sum _{r=0}^{+\infty }\text{sech}<!--\; FUNCTION\; APPLICATION\; -->(-\mathit{ry})\text{sech}<!--\; FUNCTION\; APPLICATION\; -->((-r+1)y)& \hfill & \\ \hfill & =\sum _{r=1}^{\infty }\text{sech}<!--\; FUNCTION\; APPLICATION\; -->(\mathit{ry})\text{sech}<!--\; FUNCTION\; APPLICATION\; -->((r+1)y)+\sum _{r=0}^{+\infty }\text{sech}<!--\; FUNCTION\; APPLICATION\; -->(\mathit{ry})\text{sech}<!--\; FUNCTION\; APPLICATION\; -->((r-1)y)& \hfill & \\ \hfill & =\sum _{r=1}^{\infty }\text{sech}<!--\; FUNCTION\; APPLICATION\; -->(\mathit{ry})\text{sech}<!--\; FUNCTION\; APPLICATION\; -->((r+1)y)+\sum _{r=2}^{+\infty }\text{sech}<!--\; FUNCTION\; APPLICATION\; -->(\mathit{ry})\text{sech}<!--\; FUNCTION\; APPLICATION\; -->((r-1)y)+\text{sech}<!--\; FUNCTION\; APPLICATION\; -->(y)\text{sech}<!--\; FUNCTION\; APPLICATION\; -->(0)+\text{sech}<!--\; FUNCTION\; APPLICATION\; -->(0)\text{sech}<!--\; FUNCTION\; APPLICATION\; -->(-y)& \hfill & \\ \hfill & =\sum _{r=1}^{\infty }\text{sech}<!--\; FUNCTION\; APPLICATION\; -->(\mathit{ry})\text{sech}<!--\; FUNCTION\; APPLICATION\; -->((r+1)y)+\sum _{r=1}^{+\infty }\text{sech}<!--\; FUNCTION\; APPLICATION\; -->((r+1)y)\text{sech}<!--\; FUNCTION\; APPLICATION\; -->(\mathit{ry})+2\text{sech}<!--\; FUNCTION\; APPLICATION\; -->y& \hfill & \\ \hfill & =4e^{-y}\text{cosech}<!--\; FUNCTION\; APPLICATION\; -->(2y)+2\text{sech}<!--\; FUNCTION\; APPLICATION\; -->y& \hfill & \\ \hfill & =\frac{4e^{-y}}{\text{sinh}<!--\; FUNCTION\; APPLICATION\; -->2y}+\frac{2}{\text{cosh}<!--\; FUNCTION\; APPLICATION\; -->y}& \hfill & \\ \hfill & =\frac{2e^{-y}}{\text{sinh}<!--\; FUNCTION\; APPLICATION\; -->y\text{cosh}<!--\; FUNCTION\; APPLICATION\; -->y}+\frac{2}{\text{cosh}<!--\; FUNCTION\; APPLICATION\; -->y}& \hfill & \\ \hfill & =\frac{2e^{-y}+2\text{sinh}<!--\; FUNCTION\; APPLICATION\; -->y}{\text{sinh}<!--\; FUNCTION\; APPLICATION\; -->y\text{cosh}<!--\; FUNCTION\; APPLICATION\; -->y}& \hfill & \\ \hfill & =\frac{2e^{-y}+e^y-e^{-y}}{\text{sinh}<!--\; FUNCTION\; APPLICATION\; -->y\text{cosh}<!--\; FUNCTION\; APPLICATION\; -->y}& \hfill & \\ \hfill & =\frac{e^y-e^{-y}}{\text{sinh}<!--\; FUNCTION\; APPLICATION\; -->y\text{cosh}<!--\; FUNCTION\; APPLICATION\; -->y}& \hfill & \\ \hfill & =\frac{2\text{cosh}<!--\; FUNCTION\; APPLICATION\; -->y}{\text{sinh}<!--\; FUNCTION\; APPLICATION\; -->y\text{cosh}<!--\; FUNCTION\; APPLICATION\; -->y}& \hfill & \\ \hfill & =2\text{cosech}<!--\; FUNCTION\; APPLICATION\; -->y.& \hfill & \end{array}$$