# Power Series for the Square Root of a Polynomial

In this example, we'll compute terms of the power series for √4x+1.

First, rearrange the polynomial in order of increasing powers of x. This puts the constant term first: 1+4x

If the first term is not 1, divide everything by that term, to get a polynomial whose first term is 1. Call this P.

The first term of the power series is 1. Subtract this from P to get (1+4x) - 1 = 4x.

Take the lowest-order term of that result and divide by 2 to get the
next term: 2x. Add this to the previous term to get 1+2x; square
this to get 1+4x+4x^{2}; subtract this from P to get (1+4x) -
(1+4x+4x^{2}) = -4x^{2}.

Take the lowest-order term of that result and divide by 2 to get the
next term: -2x^{2}. Add this to the previous term to get
1+2x-2x^{2}; square this to get 1+4x-8x^{3}+4x^{4}; subtract
this from P to get (1+4x) - (1+4x-8x^{3}+4x^{4}) =
8x^{3}-4x^{4}.

Take the lowest-order term of that result and divide by 2 to get the
next term: 4x^{3}. Add this to the previous term to get
1+2x-2x^{2}+4x^{3}; square this to get
1+4x+20x^{4}-16x^{5}+416x^{6}; subtract this from P to get
-20x^{4}+16x^{5}-416x^{6}.

Take the lowest-order term of that result and divide by 2 to get the
next term: -10x^{4}. Continue in a similar manner until you have the
desired number of terms.

1 + 2x - 2x^{2} + 4x^{3} - 10x^{4} + 28x^{5} - 84x^{6} +
264 x^{7} - ...

The coefficients of this series are: 1, 2, -2, 4, -10, 28, -84, 264, -858, 2860, -9724, 33592, -117572, .... It is not in Sloane's database but A2420 is the same except for the signs (and gives the coefficients of the power series for √1-4x).

This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2020 Mar 26. s.11