Acceleration of series convergence

Question

everyone! I am currently struggling with following problem: compute the series $$ \sum\limits_{m=1}^{+\infty}\frac{1}{m}\sin(m\alpha)(\cos(m\beta_1) - \cos(m\beta_2)) $$ and $$ \sum\limits_{m=1}^{+\infty}\frac{1}{m}\sin(m\alpha)(\cos(m\beta_1) - \cos(m\beta_2))\exp(-m^2\gamma) $$ Where $\alpha \approx 10^{-3}$, $\beta_{1,2} \approx 10^{-3}$ and $\gamma \approx 10^{-6}$,(thus, the parameters are very small). The main problem is in velocity of convergence of these series. Recently I have heard that there is methods of acceleration of convergence of slowly converging series. May be anyone could suggest any such technique? Thanks in advance!

Answer 1

The first series may be evaluated analytically using

$$f(\gamma)=\sum_{m=1}^{\infty} \frac{\sin{m \gamma}}{m} = \begin{cases}\\ \frac{\pi-\gamma}{2} & \gamma \in [0,2\pi)\\ f(\gamma+2 k \pi) & \gamma \in [-2k \pi,-2(k-1) \pi) \end{cases}$$

and

$$\sin{m \alpha} \cos{m \beta} = \frac12 \left [\sin{\left(m\frac{\alpha+\beta}{2}\right) }+\sin{\left(m\frac{\alpha-\beta}{2} \right)}\right ]$$

Note that the sum is not zero because $f(\gamma)$ is a sawtooth rather than a line.