We can find the Mclaurin series for an inverse function
if we have the Mclaurin series for the function
itself by expanding the inverse function in a power series with undetermined coefficient, then equating powers of
to obtain equations for the coefficients which we can solve.
Example: Find the Mclaurin series up to the term in
for
for
Assume the Mclaurin series takes the form
(1)
Since
is an odd function,
will be too, so all coefficients of even powers will be zero and
Put
so that
and (1) becomes
Equating coefficients of
Equating coefficients of
Equating coefficients of
Then
Since labels are arbitrary,