Cubic Root Expansion and Approximation

If $c{x}^3+ax-y=0$, shew that one value of $x$ is $\frac{y}{a}-\frac{c{y}^3}{a^4}+\frac{3c^2{y}^5}{a^7}-\frac{12c^3{y}^7}{a^{10}}\dots$. Hence shew that $x=0.00999999$ is an approximate solution of the equation $x^3+100x-1=0$. To how many places of decimals is the result correct?

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. XXII. Undetermined Coefficients, Examples XXII.b, No. 9