I would like to numerically compute the following series (which it related to the modified Bessel function of the second kind) up to arbitrary precision. Its definition is $$ S(x) = \sum_{k=0}^\infty \frac{1}{(k!)^2} x^k $$
Since it is fast converging it should be sufficient to check wether each summand is larger than some prescribed accuracy and then truncate the series. But summing the series starting with the largest summand seems to be a bad idea. Therefore I would save all summands in an array and after truncating the series, I sum over all summands backward. Does this make sense?
However, I wonder wether there are some more advandced techniques or acceleration tricks for numerically dealing with such series?
Summing from smallest to largest is usually a good idea. That's not necessarily "backwards", though. The largest term may be somewhere in the middle of those you want to sum. You might go from both ends towards the middle. That is, let $a$ and $b$ be the first and last indices in the range you want to sum. If $a < b$ and term $a$ < term $b$, add term $a$ to the sum and increment $a$, otherwise add term $b$ to the sum and decrement $b$. When $a=b$, add that term and stop. otherwise