. Subtract a from both sides: .
Factor an "r" out of that sum: . Divide both sides by r: .
That is almost what we started with- it is missing " " but we can add that to both sides:
Because these are finite sums, all those are legal algebraic operations and we have now arrived at . Now solve for S.
That is the sum of a finite number, n, of terms. The only place n occurs in that formula is as " ". If r> 1 that will get larger and larger without bound as n increases so we cannot "take the limit" as n increases. If r< -1, then we have alternating positive and negative values but still getting larger and larger without bound in absolute value so we cannot "take the limit" as n increases.
If -1< r< 1, then r^n gets smaller and smaller- its limit is 0 and so the limit of S is .
Finally, if r= 1 we just have S= a+ a+ a+ ....+ a = na which gets larger and larger without bound (for all a except 0) and so has no limit. If r= -1, we get S= a- a+ a- a+/cdot/cdot/cdot a which is a if n is odd and 0 if n is even- still no limit.
That limit exist only if -1< r< 1, in which case the sum is or if a= 0 in which case the sum is 0 no matter what r is.