Final answer:
The common ratio between the terms is 1/8. When the absolute value of the common ratio is less than 1, the series converges. In this case, |1/8| < 1. The given series is convergent and its value is 8/35.
Step-by-step explanation:
To determine whether the given series is convergent or divergent, we can use the concept of an infinite geometric series. The series can be expressed as:
1/5 + 1/13 + 1/21 + 1/29 + 1/37 + 1/45 + ...
The common ratio between the terms is 1/8. When the absolute value of the common ratio is less than 1, the series converges. In this case, |1/8| < 1, so the series converges.
To find its value, we can use the formula for the sum of an infinite geometric series:
Sum = a / (1 - r), where a is the first term and r is the common ratio.
Plugging in the values, we have:
Sum = (1/5) / (1 - 1/8)
Simplifying the expression, we get:
Sum = (8/40) / (7/8) = 64/280 = 8/35
Therefore, the value of the series is 8/35.