What is the smallest integer that can possibly be the sum of an infinite geometric series whose first term is 9?
To find the smallest possible sum of an infinite geometric series, we need to determine the common ratio (r) of the series.
The general formula for the sum of an infinite geometric series is given by:
S = a / (1 - r),
Where:
S = sum of the series
a = first term of the series
r = common ratio of the series
In this case, the first term (a) is given as 9. However, we need to find the smallest possible value for the common ratio (r) to get the smallest sum.
To do this, we need to consider the restrictions for a geometric series to converge. For a geometric series to converge, the absolute value of the common ratio (|r|) must be less than 1.
Since our goal is to find the smallest possible sum, we want the absolute value of the common ratio to be as close to 1 as possible without exceeding it. Therefore, we choose r = 1.
Plugging in the values into the formula, we have:
S = 9 / (1 - 1)
S = 9 / 0
However, 9 divided by 0 is undefined. Therefore, there is no smallest possible integer sum for this particular geometric series.