2017.3.8 Question 8

We have

as desired.

1. Let $a_m=1$. On one hand, we have

   $$\begin{array}{llll}\hfill \sum _{m=1}^n{a}_m(b_{m+1}-b_m)& =\sum _{m=1}^n\left[\text{sin}<!--\; FUNCTION\; APPLICATION\; -->(m+1)x-\text{sin}<!--\; FUNCTION\; APPLICATION\; -->\mathit{mx}\right]& \hfill & \\ \hfill & =\sum _{m=1}^{n}2\text{cos}<!--\; FUNCTION\; APPLICATION\; -->\left(\frac{(m+1)x+\mathit{mx}}{2}\right)\text{sin}<!--\; FUNCTION\; APPLICATION\; -->\left(\frac{(m+1)x-\mathit{mx}}{2}\right)& \hfill & \\ \hfill & =2\sum _{m=1}^n\text{cos}<!--\; FUNCTION\; APPLICATION\; -->\left(m+\frac{1}{2}\right)x\text{sin}<!--\; FUNCTION\; APPLICATION\; -->\frac{x}{2}& \hfill & \\ \hfill & =2\text{sin}<!--\; FUNCTION\; APPLICATION\; -->\frac{x}{2}\sum _{m=1}^n\text{cos}<!--\; FUNCTION\; APPLICATION\; -->\left(m+\frac{1}{2}\right)x.& \hfill & \end{array}$$

   On the other hand, we have

   $$\begin{array}{llll}\hfill \sum _{m=1}^n{a}_m(b_{m+1}-b_m)& =a_{n+1}{b}_{n+1}-a_1{b}_1-\sum _{m=1}^n{b}_{m+1}(a_{m+1}-a_m)& \hfill & \\ \hfill & =\text{sin}<!--\; FUNCTION\; APPLICATION\; -->(n+1)x-\text{sin}<!--\; FUNCTION\; APPLICATION\; -->x.& \hfill & \end{array}$$

   Therefore, by rearranging, we have

   $$\sum _{m=1}^n\text{cos}<!--\; FUNCTION\; APPLICATION\; -->\left(m+\frac{1}{2}\right)x=\frac{1}{2}\left[\text{sin}<!--\; FUNCTION\; APPLICATION\; -->(n+1)x-\text{sin}<!--\; FUNCTION\; APPLICATION\; -->x\right]\text{cosec}<!--\; FUNCTION\; APPLICATION\; -->\frac{1}{2}x$$

   as desired.

2. Let $a_m=m$, and let $b_m=\text{cos}<!--\; FUNCTION\; APPLICATION\; -->\left(m-\frac{1}{2}\right)x$. We have the identity

   $$\text{cos}<!--\; FUNCTION\; APPLICATION\; -->A-\text{cos}<!--\; FUNCTION\; APPLICATION\; -->B=-2\text{sin}<!--\; FUNCTION\; APPLICATION\; -->\left(\frac{A+B}{2}\right)\text{sin}<!--\; FUNCTION\; APPLICATION\; -->\left(\frac{A-B}{2}\right).$$

   Therefore, we have

   $$\begin{array}{llll}\hfill \sum _{m=1}^n{a}_m(b_{m+1}-b_m)& =\sum _{m=1}^{n}m\cdot \left[\text{cos}<!--\; FUNCTION\; APPLICATION\; -->\left(m+\frac{1}{2}\right)x-\text{cos}<!--\; FUNCTION\; APPLICATION\; -->\left(m-\frac{1}{2}\right)x\right]& \hfill & \\ \hfill & =\sum _{m=1}^n-2m\text{sin}<!--\; FUNCTION\; APPLICATION\; -->\mathit{mx}\text{sin}<!--\; FUNCTION\; APPLICATION\; -->\frac{1}{2}x& \hfill & \\ \hfill & =-2\text{sin}<!--\; FUNCTION\; APPLICATION\; -->\frac{1}{2}x\sum _{m=1}^{n}m\text{sin}<!--\; FUNCTION\; APPLICATION\; -->\mathit{mx},& \hfill & \end{array}$$

   and

   $$\begin{array}{llll}\hfill & \sum _{m=1}^n{a}_m(b_{m+1}-b_m)& \hfill & \\ \hfill & =a_{n+1}{b}_{n+1}-a_1{b}_1-\sum _{m=1}^n{b}_{m+1}(a_{m+1}-a_m)& \hfill & \\ \hfill & =(n+1)\text{cos}<!--\; FUNCTION\; APPLICATION\; -->\left(n+\frac{1}{2}\right)x-1\cdot \text{cos}<!--\; FUNCTION\; APPLICATION\; -->\frac{1}{2}x-\sum _{m=1}^n\text{cos}<!--\; FUNCTION\; APPLICATION\; -->\left(m+\frac{1}{2}\right)x\cdot 1& \hfill & \\ \hfill & =(n+1)\text{cos}<!--\; FUNCTION\; APPLICATION\; -->\left(n+\frac{1}{2}\right)x-\text{cos}<!--\; FUNCTION\; APPLICATION\; -->\frac{1}{2}x-\sum _{m=1}^n\text{cos}<!--\; FUNCTION\; APPLICATION\; -->\left(m+\frac{1}{2}\right)x& \hfill & \\ \hfill & =(n+1)\text{cos}<!--\; FUNCTION\; APPLICATION\; -->\left(n+\frac{1}{2}\right)x-\text{cos}<!--\; FUNCTION\; APPLICATION\; -->\frac{1}{2}x-\frac{1}{2}\left(\text{sin}<!--\; FUNCTION\; APPLICATION\; -->(n+1)x-\text{sin}<!--\; FUNCTION\; APPLICATION\; -->x\right)\text{cosec}<!--\; FUNCTION\; APPLICATION\; -->\frac{1}{2}x& \hfill & \\ \hfill & =\frac{1}{2}\text{cosec}<!--\; FUNCTION\; APPLICATION\; -->\frac{1}{2}x\left[2(n+1)\text{cos}<!--\; FUNCTION\; APPLICATION\; -->\left(n+\frac{1}{2}\right)x\text{sin}<!--\; FUNCTION\; APPLICATION\; -->\frac{1}{2}x-2\text{cos}<!--\; FUNCTION\; APPLICATION\; -->\frac{1}{2}x\text{sin}<!--\; FUNCTION\; APPLICATION\; -->\frac{1}{2}x-\left(\text{sin}<!--\; FUNCTION\; APPLICATION\; -->(n+1)x-\text{sin}<!--\; FUNCTION\; APPLICATION\; -->x\right)\right]& \hfill & \\ \hfill & =\frac{1}{2}\text{cosec}<!--\; FUNCTION\; APPLICATION\; -->\frac{1}{2}x\left[(n+1)\left(\text{sin}<!--\; FUNCTION\; APPLICATION\; -->(n+1)x-\text{sin}<!--\; FUNCTION\; APPLICATION\; -->\mathit{nx}\right)-(\text{sin}<!--\; FUNCTION\; APPLICATION\; -->x-\text{sin}<!--\; FUNCTION\; APPLICATION\; -->0)-\left(\text{sin}<!--\; FUNCTION\; APPLICATION\; -->(n+1)x-\text{sin}<!--\; FUNCTION\; APPLICATION\; -->x\right)\right]& \hfill & \\ \hfill & =\frac{1}{2}\text{cosec}<!--\; FUNCTION\; APPLICATION\; -->\frac{1}{2}x\left[n\text{sin}<!--\; FUNCTION\; APPLICATION\; -->(n+1)x-(n+1)\text{sin}<!--\; FUNCTION\; APPLICATION\; -->\mathit{nx}\right].& \hfill & \end{array}$$

   Therefore, we have

   $$\begin{array}{llll}\hfill -2\text{sin}<!--\; FUNCTION\; APPLICATION\; -->\frac{1}{2}x\sum _{m=1}^{n}m\text{sin}<!--\; FUNCTION\; APPLICATION\; -->\mathit{mx}& =\frac{1}{2}\text{cosec}<!--\; FUNCTION\; APPLICATION\; -->\frac{1}{2}x\left[n\text{sin}<!--\; FUNCTION\; APPLICATION\; -->(n+1)x-(n+1)\text{sin}<!--\; FUNCTION\; APPLICATION\; -->\mathit{nx}\right]& \hfill & \\ \hfill \sum _{m=1}^{n}m\text{sin}<!--\; FUNCTION\; APPLICATION\; -->\mathit{mx}& =-\frac{1}{4}{\text{cosec}<!--\; FUNCTION\; APPLICATION\; -->}^2\frac{1}{2}x\left[n\text{sin}<!--\; FUNCTION\; APPLICATION\; -->(n+1)x-(n+1)\text{sin}<!--\; FUNCTION\; APPLICATION\; -->\mathit{nx}\right],& \hfill & \end{array}$$

   and therefore, $p=-\frac{1}{4}n$, $q=\frac{1}{4}(n+1)$.