Algorithm for finding the minimum index in array with value exceeding x
Assuming the $s_i$ positive (which wasn't said in the OP), the prefix sum of $s_i$ forms a monotonic sequence.

You can start a simultaneous exponential search from both ends, followed by a standard binary search when the half sum has been crossed. The left exponential search takes $O(\log(j))$ and the right one $O(\log(k-j))$. As they are performed simultaneously, the fastest "wins".