Finding Mclaurin Series for an Inverse Function

We can find the Mclaurin series for an inverse functionif we have the Mclaurin series for the functionitself by expanding the inverse function in a power series with undetermined coefficient, then equating powers ofto obtain equations for the coefficients which we can solve.

Example: Find the Mclaurin series up to the term infor


Assume the Mclaurin series takes the form(1)

Sinceis an odd function,will be too, so all coefficients of even powers will be zero and

Putso thatand (1) becomes

Equating coefficients of

Equating coefficients of

Equating coefficients of


Since labels are arbitrary,

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