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Since $a_1=f^{\prime }(0)\ne 0$, substitute the proposed inverse series into the forward one. Comparing coefficients of $f,f^2,f^3$ givesSolving successively yields the cubic series reversion coefficientsThe conditions $f(0)=0$ and $a_1\ne 0$ ensure that both local series are centred at zero and that the inverse has a linear leading term.

Series reversion constructs the power series for a local inverse function. If $f(g)=a_{1}g+a_2{g}^2+\cdots$ with $a_1\ne 0$, substitute $g=b_{1}f+b_2{f}^2+\cdots$ into the original power series and compare coefficients. The linear coefficient determines $b_1$; each later coefficient enters linearly after the earlier ones are known. This also defines reversion of a formal power series, independently of convergence.

For example, $f=g+g^2$ givesSubstitution makes the coefficients of $f^2$ and $f^3$ cancel. The exact local inverse $g=(\sqrt{1+4f}-1)/2$ verifies the calculation. The general coefficients through cubic order are derived under cubic series reversion.

For an analytic function with nonzero derivative, the inverse function theorem supplies an analytic local inverse, so the reverted series converges sufficiently near the base point. If the linear coefficient vanishes, an ordinary inverse power series may fail: $f=g^2$ instead needs a choice of square-root branch and a fractional scale.