text: sumation notation n=1 to infinity of {(-1)^n*(x+6)^n}/n*6^n..... find radius of convergence? find the interval odf convergence use the ratio test. limit as n goes to infinity of {[(x+6)^(n+1)]/[(n+1)6^(n+1)]}/{[(x+6)^n]/[n*6^n]} which simplifies to limit as n goes to infinity of [n(x+6)]/[(n+1)*6] = (x+6)/6 set the absolute value of this to be less than 1 |(x+6)/6|<1 thus, the radius of convergence is 6. to find the interval, solve for x |(x+6)/6|<1 -1 < (x+6)/6 < 1  multiply everything by 6 -6 < x+6 < 6  subtract 6 from everything -12 < x < 0 the endpoints may or may not be included.  they have to be checked.  if you do the check, you will find that it converges at 0 but not -12.  thus, the interval of convergence is (-12,0]