I want to take partial sums of a convergent series. But how will I know how accurate the approximation will be? I am looking for practical ways of achieving this even if not very rigorous!
Suppose the terms of the series are given by $a_n$, its partial sums by $s_n$ and a desired accuracy of $\epsilon$ is required. Keeping questions of stability aside for now how can I achieve this?
I have seen termination criteria such as;
- keep summing (i.e. constructing partial sums) until $a_n
. - keep summing (i.e. constructing partial sums) until $\frac{s_n-s_{n-1}}{s_{n-1}}<\epsilon$.
But what is the justification behind these termination criterion??
Can Cauchy's criteria for convergence be used in some way?
I'm asking because I have a taylor series for some function, but trying to put a bound on the higher order derivatives is next to futile so Cauchy's and Lagrange's form of the remainder are not an option. Even though the function is strictly monotone (hence I know the maximum occurs at the endpoints), but cauchy's estimate requires one to know the maximum modulus on a closed disk. And I have no idea how to do this since I dont know the function in closed form. Hence a bound for the remainder through cauchy's estimate seems out the window also.
The series is found through a solution of a nonlinear ODE. Hence the coefficients are given recursively, so I don't know how I could apply the geometric series or integral tests to find an answer.
Surely people have faced similar problems so I was wondering what one would do in practice.