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

`\frac1{1-x}=\displaystyle\sum_(n=0)^\infty x^n`

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

`f(x)=\frac4{1-2x}=\displaystyle4\sum_(n=0)^\infty(2x)^n`

Take the derivative:

`f'(x)=\frac8{(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|((n+2)(2x)^(n+1))/((n+1)(2x)^n)\right|=|2x|\lim_(n\to\infty)(n+2)/(n+1)=|2x|<1`

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