Interval of convergence

Question

Lets say you have a series ∑_n=1^∞ ((3x+2)^(n+1))/n^(5/2). What would the interval of convergence be?

Answer 1

Using the ratio test, we have

limit abs( (3x+2)⁽ⁿ⁺²⁾*n^(5/2)/(3x+2)⁽ⁿ⁺¹⁾*(n+1)^{(5/2) )}

n>infinity

= limit abs( (3x+2)*n^(5/2)/(n+1)^(5/2) ) = abs( (3x+2)*1 )

n> infinity

Since the absolute value must be less than 1 to converge, we have abs( 3x+2 ) < 1

or -1 < 3x+2 < 1 or -3< 3x < -1 or -1 < x < -(1/3) the interval of convergence