Simplify a finite series

Question

can someone help me simplify the following formula? $$y = \pi(x-a)-\frac{(\pi(x-a))^3}{3!}+\frac{(\pi(x-a))^5}{5!}-\frac{(\pi(x-a))^7}{7!}+\frac{(\pi(x-a))^9}{9!}-\frac{(\pi(x-a))^{11}}{11!}+\frac{(\pi(x-a))^{13}}{13!}-\frac{(\pi(x-a))^{15}}{15!}+\frac{(\pi(x-a))^{17}}{17!}-\frac{(\pi(x-a))^{19}}{19!}+\frac{(\pi(x-a))^{21}}{21!}-\frac{(\pi(x-a))^{23}}{23!}$$

Answer 1

If you work with this limited expression, first let $t=\pi(x-a)$ to write $$y=\sum_{i=0}^{11}(-1)^n\frac {t^{2n+1}}{(2n+1)!}=t\sum_{i=0}^{11}(-1)^n\frac {t^{2n}}{(2n+1)!}$$ You have the obvious root $t=0$. For the other roots, let $t^2=u$ and then consider solving for $u$ the polynomial equation $$\sum_{i=0}^{11}(-1)^n\frac {u^{n}}{(2n+1)!}=0$$ which is not most pleasant task. There are $11$ roots but only three of them are real. Graphing, they look to be very close to $(10,40,85)$

In my opinion, the easiest way to get the accurate solutions is to use Newton method starting from these estimates. The iterates would be $$\left( \begin{array}{cc} n & u_n \\ 0 & 10.00000000 \\ 1 & 9.868296250 \\ 2 & 9.869604271 \\ 3 & 9.869604401 \end{array} \right)$$

$$\left( \begin{array}{cc} n & u_n \\ 0 & 40.00000000 \\ 1 & 39.47302741 \\ 2 & 39.47848608 \\ 3 & 39.47848665 \end{array} \right)$$ $$\left( \begin{array}{cc} n & u_n \\ 0 & 85.00000000 \\ 1 & 86.98238456 \\ 2 & 86.95292238 \\ 3 & 86.95291554 \end{array} \right)$$ Then the six non-trivial solutions for $t$ (they are quite close to $\pm \pi$, $\pm 2\pi$, $\pm 3\pi$).