I'm trying to estimate the parameters of a Hawkes process using an EM algorithm. I found this paper where in the maximisation step, for parameter $\mu$, we have $$\mu^{k+1} = \frac{\sum^n_{i=1}p^k_{ii}}{T},$$ where $$p_{ii}^k=\frac{\mu^k}{\mu^k+\sum^{i-1}_{j=1}\alpha^k\beta^k e^{-\beta^k(t_i-t_j)}}.$$ This is achieved by setting the partial derivative of the log-likelihood $$\frac{\partial}{\partial\mu}l=\sum^n_{i=1} \frac{1}{\lambda_i(t_i)}-T,$$ where $$\lambda_i(t_i) = \mu+\sum^{i-1}_{j=1}\alpha\beta e^{-\beta(t_i-t_j)}$$ to zero and rewriting it in terms of $\mu$. However, I don't see how we arrive to this. Here's what I've tried so far:
$$\sum^n_{i=1} \frac{1}{\mu+\sum^{k-1}_{j=1}\alpha\beta e^{-\beta(t_i-t_j)}}-T=0 \\ \sum^n_{i=1} \frac{1}{\mu+\sum^{k-1}_{j=1}\alpha\beta e^{-\beta(t_i-t_j)}}=T.$$
However, I'm stuck here due to the summations.