If x+3 and x-2 are the first two terms of a geometric series, find the values of x for which the series converges

Answer: when , the series will converge.sa1======since this is a geometric ratio, figure out the common ratio.the common ratio is the ratio between a term and the term before it. divide x-2 by the term before it, which is x+3.    for a geometric series to be convergent, the common ratio r must be between -1 and 1, or  .    we can multiply both sides by  |x+3| because  |x+3| is always positive. there are no inequality flips.    these absolute values represented in piecewise format:     and    consider all possibilities. the left of x=-3 (and at x=3), between x=-3 and x=2, and the right of x=2 (and at x=2)  are areas to check for since they change the signs of the piecewise expressions.case 1: when , then      and  therefore    the last step was achieved by adding x to both sides.this is always a false statement. we obtained no solution here.case 2: when      and  therefore    but this case is only for  the intersection between  and  is      so this is one of the solutions.case 3: when  , then    and  therefore    this is always a true statement.therefore,  is another solution.======the solution from case 2 is  the solution from case 3 is  both of these solutions will work, so we can just combine them.joining these solutions together, we can simply say that  . this will cover the solutions from cases 2 and 3.so when  , the series will converge.