Acceleration of series convergence 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!

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.  
