SOLUTION: What is the smallest integer that can possibly be the sum of an infinite geometric series whose first term is 9?

Question 1036781: What is the smallest integer that can possibly be the sum of an infinite geometric series whose first term is 9?
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
If r = 0, then |r| <= 1 is certainly true making

S = a/(1-r) = 9/(1-0) = 9/1 = 9

So the smallest possible infinite sum is 9. This is where the first term is 9 and the successive terms after the first term are all 0.

Ie

S = 9+0+0+0+0+0+.... = 9

we're adding infinitely many 0s to the first term 9

If 0 < r < 1, then S will be larger than 9. For instance, if r = 0.1, then

S = a/(1-r) = 9/(1-0.1) = 9/0.9 = 10

so we can see that the sum is larger if r > 0.

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So again, to summarize, if the common ratio is r = 0 then the infinite sum is 9.
This is the smallest infinite sum possible.
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