d^2/dx^2 (1 - 2x)^-1
The coefficient of x^5 in the inverse of (1 - 2x)^-1 in R[[x]] is 2^5 = 32. R[[x]] is the set of power series a(0) + a(1) x^1 + a(2) x^2 + . . . with a(i)'s real numbers. Dividing 1 by (1 - 2x) in a long division will convince the reader of this fact. What is the coefficient of x^5 in the second derivative of the inverse of (1 - 2x)^-1?
The answer is 5376.
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1 solution
1 Divide by (1 - 2y) in a long division gives the power serie 1 + 2^1 x y^1 + 2^2 x y^2 + 2^3 x y^3 + 2^4 x y^4 + 2^5 x y^5 + 2^6 x y^6 + 2^7 x y^7 + . . .. The x here is the multiplication sign. The coefficient of x^5 in the second derivative of this power serie is (2^7) x 7 x 6 = 5376.