What is the smallest integer that can possibly be the sum of an infinite geometric series whose first term is 1?
S = 1/(1-r)
You need |r| < 1, so what do you think?
What's the smallest positive integer that can possibly be the sum of an infinite geometric series whose first term is 1?
To find the smallest integer that can possibly be the sum of an infinite geometric series, we need to consider the formula for the sum of an infinite geometric series:
S = a / (1 - r)
Where:
S is the sum of the infinite series,
a is the first term of the series, and
r is the common ratio of the series.
In this case, the first term (a) is given as 1. However, since we are looking for an integer sum, we need to find a common ratio (r) that will result in an integer value for S.
To find such a ratio, we need to consider some properties of geometric series. For an infinite geometric series to converge (meaning the sum does not go to infinity), the absolute value of the common ratio (|r|) must be less than 1.
To find the smallest integer sum, we need to find the smallest possible |r| that satisfies this condition.
Let's try out various values for |r| and calculate the corresponding sum:
For |r| = 1/2:
S = 1 / (1 - 1/2) = 1 / (1/2) = 2
For |r| = 1/3:
S = 1 / (1 - 1/3) = 1 / (2/3) = 3/2 = 1.5 (not an integer)
For |r| = 1/4:
S = 1 / (1 - 1/4) = 1 / (3/4) = 4/3 (not an integer)
We can keep trying smaller values for |r|, but we notice that as |r| approaches zero, the sum becomes larger. Since we are looking for the smallest possible integer sum, we need to find the largest possible value of |r| that still satisfies the condition |r| < 1.
Therefore, the smallest integer sum will occur when |r| is the largest possible value less than 1. In this case, |r| = 1 - ε, where ε is an infinitesimally small positive number.
Using this value, the sum S becomes:
S = 1 / (1 - (1 - ε)) = 1 / ε
Since ε is infinitesimally small, the reciprocal of ε becomes infinitely large. Therefore, the smallest integer sum of the geometric series is infinity.
In conclusion, the smallest integer that can possibly be the sum of an infinite geometric series whose first term is 1 is infinity.