you could actually determine the series's progression and know it's common term and write down as $\sum_{0}^{n}$f(x) {i.e a common term} and determine the summation for fewer problems(like those expansion of e! and log(1+x) ) and for others what you are doing is right because for few problems there is no other way except for writing out free sums ........ and when you practice it would help you a lot since few problems are interconnection in pattern and grouping the variables ....... YOU NEED TO REFER FEW BOOKS TO PRACTICE THOSE PROBLEMS FOR QUICKER ANSWERING....BUT PRACTICE TIME WOULDN'T TAKE MUCH OF YOUR TIME