Error estimate of definite integral of a taylor expanded function

Question

If I consider a monotonic decreasing function $f(x)$ in the interval $[0,+\infty[$, and I consider the definite integral $\int_{0}^{+\infty}f(x)\,\mathrm{d}x$. What is the error committed if I compute the definite integral of the taylor expansion of $f(x)$ around $x=0$ truncate at a certain order $n$? Thanks in advance

Answer 1

The mistake is that if you take $T_n$ which is the $n$-th Taylor polynomial for $f$ around $x$, then

$$\int_0^\infty T_n(x) dx$$ is either $\infty$ (if the leading coefficient is positive) or $-\infty$ (if the leading coefficient is negative). So unless $$\int_0^\infty f(x)dx$$ is also infinite, the error commited is infinite.