  In G.I. Taylor's historic paper on swimming microorganisms, a two dimensional
sheet was proposed as a model for flagellated cells passing traveling waves as
a means of locomotion. Using a perturbation series, Taylor computed swimming
speeds up to fourth order in amplitude. Here we systematize that expansion so
that it can be carried out formally to arbitrarily high order. The resultant
series diverges for an order one value of the wave amplitude, but may be
transformed into series with much improved convergence properties and which
yield results comparing favorably to those obtained numerically via a boundary
integral method for moderate and large values of the wave amplitudes.
