Borel transform technique and then-bubble-diagram contribution to the lepton anomaly

By using the Borel transform technique we calculate analytically the muon anomaly from the mass-dependent $n$-bubble diagram in the limit where the mass ratio $\frac{m_{\mu }}{m_e}$ is large. For large $n$ the sign oscillates, in contrast with the correct anomaly. The analysis shows that the validity of expanding in $L\equiv \ln \left(\frac{m_{\mu }}{m_e}\right)$ is strongly dependent on the order $N=2n+2\mathrm{}$, in which we calculate.