Question
What is the convergence set for
Explanation / Answer
Using the root test we get lim(n->infinity) |(x^(n+1))/((5^(n+1))((n+1)+1)^3) * ((5^n)(n+1)^3)/x^n| = lim(n->infinity) |(x/5)*((n+1)/(n+2))^3| = |x/5| For this to converge, we need |x/5| < 1 So that implies |x| < 5 or x is in (-5,5). Now we must test the endpoints. If x = 5, the series turns into the sum of 1/(n+1)^3, which converges by p-series. If x = -5, the series turns into the sum of (-1)^n / (n+1)^3, which also converges by alternating series test. Therefore the convergence set would be [-5,5].