Question

0 Question # 3. (3 marks) Consider the power series, f(x) = 3 an(x + 1)". Suppose we know that f(-4), as a series, diveryes, while (2) converges. Determine the radius of convergence of the power series for f'(x). Precisely name the results we learned in Week 3 that you use, and where you are using them.

Answer 1

The given series is n f(x) = { an (x+1) nzo This is a Power series about x = - 1 Now we know that if radious of convergent is r the the interval of convergence is (-1-2, - 1+r). and the series diverges in (-2,-1-2) u fltr, a) Again we have FC4) a series diverges ie f(-1-3) as a sesies diverses as and FCO) = f(1+3) OLS a series converges. Henee the radious of converges ise of the Power series is 3. Again we know that if & is a radious of convergence representation of f(n) then t'm, is so. of a power series, Hence radious of convergence of power series for les is also 3.