Question

Which of the following real numbers can be expressed as the sum of an infinite geometric series?

a) 1/2
b) 3/4
c) 5/3
d) 4/7

Answer 1

The real numbers 1/2, 3/4, and 4/7 can be expressed as the sum of an infinite geometric series because they fit the criteria for the sum to converge. The number 5/3 cannot be expressed as a convergent infinite geometric series because it is greater than 1.

Step-by-step explanation:

The question relates to whether certain real numbers can be expressed as the sum of an infinite geometric series. A series is geometric if there is a common ratio between successive terms. An infinite geometric series converges to a sum only if the absolute value of the common ratio is less than 1. When it does converge, the sum can be found using the formula S = a / (1 - r), where 'a' is the first term and 'r' is the common ratio of the series.

Let's consider the options given:

• 1/2 can be expressed as the sum of an infinite geometric series with a = 1 and r = 1/2.
• 3/4 can be expressed as the sum of an infinite geometric series with a = 1/2 and r = 1/2.
• 5/3 cannot be expressed as the sum of an infinite geometric series that converges because it is greater than 1. For a geometric series to converge, the common ratio must be between -1 and 1, exclusive.
• 4/7 can be expressed as the sum of an infinite geometric series with a suitable choice of 'a' and 'r' within the constraints.

Therefore, the real numbers 1/2, 3/4, and 4/7 can be expressed as the sum of an infinite geometric series, while 5/3 cannot.