Suppose , $$\sum_{s=0}^{\infty} a_sx^s$$
is a convergent power series, but very slowly converging. In the case, the power series has a closed form, this is no problem,
$$\sum_{s=0}^\infty 0.999999^s$$ can easily be calculated because of $$\sum_{s=0}^\infty x^s=\frac{1}{1-x}$$ for $-1<x<1$.
But what, if we do not have such a closed form ?
Is it always possible to find the limit with reasonable accuracy, even if the series converges awfully slowly ?
Or can it be very difficult (or even infeasible in practice) to calculate the sum ?
We can assume that the sum is small enough for a computer program to handle it. (Of course, the value could be somewhere near $10^{10^{100}}$).
I know that there are methods to accelerate the convergence, but can we be sure that they actually give a result near the actual limit ?
You may use series acceleration techniques, like Wynn's epsilon method for linearly convergent series or even accelerate logarithmicaly convergent series. These methods can also give error estimators, which can be used to construct reliable stopping rules.
You may look at Weniger, Ernst Joachim. "Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series." Computer Physics Reports 10.5-6 (1989): 189-371. This paper analyses different algorithms and provide error analysis.