Let $f(x)$ be a smooth function that takes both positive and negative values and suppose there exists an increasing sequence of positive numbers $R_i$ diverging to $\infty$ such that $$\lim_{i \uparrow \infty}\int\limits_0^{R_i} f(x) e^{-x/R_i} dx = A > 0, $$ $$\lim_{i \uparrow \infty}\int\limits_0^{R_i} f(x) \ dx = B > 0. $$
Does it follow that $$\limsup_{i \uparrow \infty}\int\limits_0^{R_i} f(x) \left[1 - \frac{x}{R_i} + \frac{x^2}{2R_i^2}\right] \ dx > -\infty? $$
Otherwise, can one infer this conclusion when the second order polynomial in the integrand is replaced by a sufficiently high order truncation of the Taylor series expansion of $e^{-x/R_i}$ (that is to say, when the quadratic function in the integrand is replaced by the $N$-th degree polynomial approximation of the exponential function, where $N$ may be as large as we please, but is fixed)?
This problem arose in a study of summability methods for integrals.
PS: I asked this question on Stack Exchange, where it has not been answered.
