Given the potential $\phi = \alpha/r^n$, I want to find the second virial coefficient of my system. My integration limits are from some cut-off length $D$ because the particles don't experience a self-force. This was my procedure: $$B_2(T) = -2\pi \int_{0}^{\infty} (e^{-\beta \alpha /r^n} -1) r^2 dr \\ = -2\pi \int_{0}^{\infty} r^2\sum_{k=1}^{\infty} (-1)^k \frac{(\beta \alpha)^k}{r^{kn} k!} dr \\ = -2\pi \int_{0}^{\infty} \sum_{k=1}^{\infty} (-1)^k \frac{(\beta \alpha)^kr^{-kn+2}}{ k!} dr \\ = -2\pi \sum_{k=1}^{\infty} \int_{D}^{\infty} (-1)^k \frac{(\beta \alpha)^kr^{-kn+2}}{ k!} dr \\ = -2\pi \sum_{k=1}^{\infty}(-1)^k \frac{(\beta \alpha)^k}{ k!} \int_{D}^{\infty} r^{-kn+2}dr \\ = -2\pi \sum_{k=1}^{\infty}(-1)^k \frac{(\beta \alpha)^k}{ k!} \cdot \frac{r^{-kn+3}}{-kn+3} \bigg|_{D} ^{\infty} \\ = 2\pi \sum_{k=1}^{\infty}(-1)^k \frac{(\beta \alpha)^k}{ k!} \cdot \frac{D^{-kn+3}}{-kn+3}$$
My question is, is there a well-defined function with a Maclaurin series defined as the last line?