Mmmm.......I not certain that the "guest's" answer is correct...if "r" is > 1 or < -1, the series diverges and has no sum
Note:
9 / [ 1 - ( - 1/2)] = 6
9 / [ 1 - (- 4/5 )] = 5
So....we are trying to find some "r" between -1 and 1 such that
9 / [ 1 - r) ] = 4 → r = -5/4 ......this series will diverge
And
9/ [ 1 - r ] = 3 → r = -2.........and this series will diverge, as well
Seemingly, for the positive integer sums of less than 5, "r" will be less than -1....and any such series will diverge
So...it appears that the smallest integer that can be the sum of an infinite series whose first term is 9 is produced when r = -4/5.....and that sum is 5
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