Optimal truncation

Remarks

Some authors include the smallest term, or half of it; these choices change the error only by a fraction of the smallest term. The index $N(x)$ depends on $x$ and usually tends to infinity as $x \to x_0$.

Example: $F(x) = \int_0^\infty \frac{e^{-t}}{1 + xt}\,dt \sim \sum_{n \ge 0} (-1)^n n!\, x^n$ as $x \to 0^+$. For $x = 0.1$ the terms $n!\, (0.1)^n$ decrease while $n + 1 < 10$, the terms with $n = 9$ and $n = 10$ are equal, and then they increase. So $N(0.1) = 9$, and the error of the optimally truncated sum is at most the first omitted term $9!\, (0.1)^9 \approx 3.6 \cdot 10^{-4}$, since the remainder after $N$ terms is bounded by the next term.

Non-example: for a convergent series such as $\frac{1}{1 - x} = \sum x^n$ at $x = \frac12$ the terms decrease forever, no minimum is attained, and optimal truncation is not defined; one simply sums the series.