near the pole. For example, one can deform the integration contour in (\[borel-inverse\]) into the complex plane so that it runs above the pole or below the pole, or one can take a linear combination of these two contours, such as the principal value. These prescriptions all differ by amounts that are proportional to the residue of the integrand of (\[borel-inverse\]) at the pole. If the pole in $\widetilde{f} (t)$ is at the point $t^*=u^*/\beta_0$ and has residue $R^*$, the ambiguity in $f(\alpha_s)$ has the form $$\Delta f (\alpha_s) \;=\; K\, (2 \pi \,R^* )
      \, e^{- { t^* / \alpha_s(\mu^2)  } } \;,
\label{ambigu}$$ where $K$ is a constant of order unity. In $\alpha_s(\mu^2)$, the scale $\mu$ is typically chosen to be the largest scale in the physical process. For large $\mu$ $$\alpha_s (\mu) \approx
{1 \over \beta_0 \ln ({\mu^2 / \Lambda_{\rm QCD} ^2 }) }.
\label{alphas}$$ Inserting this expression into (\[ambigu\]), we see that the renormalon ambiguity is given approximately by $$\Delta f \;\approx \; K ( 2 \pi \,R^*)\,\left({\Lambda_{\rm QCD} ^2
\over \mu^2 }  \right)^{u^*} \;.
\label{ambigu:2}$$ Thus, the ambiguity is suppressed as a power of $\mu^2$. However, such an ambiguity can be of practical importance, since 