Sum of a Power SeriesDate: 02/10/2001 at 18:40:29 From: Ivan Subject: The sum of a power series Can you tell me how to calculate the sum of this power series? x + 4x^2 + 9x^3 + 16x^4 + ... + n^2x^n + ... |x|<1 I've found that the series is convergent, but can't get the sum. Thanks, Ivan Date: 02/10/2001 at 19:21:12 From: Doctor Anthony Subject: Re: The sum of a power series Multiply the original series by x: S = x + 4x^2 + 9x^3 + 16x^4 + ... + n^2x^n + ... xS = x^2 + 4x^3 + 9x^4 + ... + (n-1)^2x^n + ... Subtract the second series above from the first: (1-x)S = x + 3x^2 + 5x^3 + 7x^4 + ... + (2n-1)x^n + ... x(1-x)S = x^2 + 3x^3 + 5x^4 + ... + (2n-3)x^n + ... Repeat: (1-x)^2*S = x + 2x^2 + 2x^3 + 2x^4 + ... = x + 2x^2[1 + x + x^2 + x^3 + ... Replace the series in brackets with 1/(1-x): (1-x)^2*S = x + 2x^2/(1-x) x 2x^2 S = ------- + ------- (1-x)^2 (1-x)^3 - Doctor Anthony, The Math Forum http://mathforum.org/dr.math/ |
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