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+4x2; subtract this from P to get (1+4x) - (1+4x+4x2) = -4x2.

Take the lowest-order term of that result and divide by 2 to get the next term: -2x2. Add this to the previous term to get 1+2x-2x2; square this to get 1+4x-8x3+4x4; subtract this from P to get (1+4x) - (1+4x-8x3+4x4) = 8x3-4x4.

Take the lowest-order term of that result and divide by 2 to get the next term: 4x3. Add this to the previous term to get 1+2x-2x2+4x3; square this to get 1+4x+20x4-16x5+416x6; subtract this from P to get -20x4+16x5-416x6.

Take the lowest-order term of that result and divide by 2 to get the next term: -10x4. Continue in a similar manner until you have the desired number of terms.

1 + 2x - 2x2 + 4x3 - 10x4 + 28x5 - 84x6 + 264 x7 - ...

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).


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