Question

xpress 8/(1 - 2x)2 as a power series by differentiating the equation below. What is the radius of convergence? 4 (1 - 2x) = 4(1 + 2x + 4x2 + 8x3 + ...) = 4 [infinity] Σ n=0 (2x)n SOLUTION Differentiating each side of the equation, we get 8 (1 - 2x)2 = 4(2 + Correct: Your answer is correct. + 24x2 + ...) = 4 [infinity] Σ n=1 Incorrect: Your answer is incorrect. If we wish, we can replace

Answer 1

Recall that for |x| < 1, we have

$\dfrac1{1-x}=\displaystyle\sum_{n=0}^\infty x^n$

Replace x with 2x, multiply 4, and call this function f :

$f(x)=\dfrac4{1-2x}=\displaystyle4\sum_{n=0}^\infty(2x)^n$

Take the derivative:

$f'(x)=\dfrac8{(1-2x)^2}=\displaystyle8\sum_{n=0}^\infty n(2x)^{n-1}=\boxed{8\sum_{n=0}^\infty (n+1)(2x)^n}$

By the ratio test, the series converges for

$\displaystyle\lim_{n\to\infty}\left|\frac{(n+2)(2x)^{n+1}}{(n+1)(2x)^n}\right|=|2x|\lim_{n\to\infty}\frac{n+2}{n+1}=|2x|$

or |x| < 1/2, so the radius of convergence is 1/2.