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 sum of an infinite geometric series with a first term of 9, we need to determine the common ratio. Let's denote the common ratio as 'r'.
The formula for the sum of an infinite geometric series is given by:
S = a / (1 - r)
where 'S' is the sum and 'a' is the first term. In this case, 'a' is 9.
Now, we need to find the value of 'r' that results in the smallest sum. Since the common ratio must be between -1 and 1 for the series to converge, we know that |r| < 1.
To find the smallest possible sum, we want the denominator (1 - r) to be as close to zero as possible without actually reaching zero. Thus, we want the value of 'r' that is closest to -1.
Therefore, the smallest possible sum occurs when 'r' is -1.
Plugging in the values of 'a' and 'r', we have:
S = 9 / (1 - (-1))
= 9 / (1 + 1)
= 9 / 2
= 4.5
So, the smallest integer that can be the sum of an infinite geometric series with a first term of 9 is 4.5.