I am studying exponential matrices , and there is a part of a proof that I don't understand .
Given the matrix $$ J=\left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right) $$ the power of $J$ are eas to compute $$ J^{2 k}=(-1)^k I, J^{2 k+1}=(-1)^k J $$ and so $$ \begin{aligned} e^{\beta_j J t} & =\sum_{k=0}^{+\infty} \frac{\beta_j^k J^k t^k}{k !} \\ & =\left(\sum_{k=0}^{+\infty} \frac{(-1)^k\left(\beta_j t\right)^{2 k}}{(2 k) !}\right) I+\left(\sum_{k=0}^{+\infty} \frac{(-1)^k\left(\beta_j t\right)^{2 k+1}}{(2 k+1) !}\right) J \\ & =\cos \left(\beta_j t\right) I+\sin \left(\beta_j t\right) J=\left(\begin{array}{cc} \cos \left(\beta_j t\right) & \sin \left(\beta_j t\right) \\ -\sin \left(\beta_j t\right) & \cos \left(\beta_j t\right) \end{array}\right) \end{aligned} $$
The part that I don't understand is when he separate the infinite summation in 2 parts . I Don't know why is possible to do that ,in general you can't separate an infinite series .