To determine whether an infinite geometric series converges or diverges, we need to examine the common ratio (r) between terms. In this case, we can see that each term is obtained by multiplying the previous term by a common ratio of 1/5.
An infinite geometric series converges if the absolute value of the common ratio is less than 1 (|r| < 1). In this case, |1/5| = 1/5 and since 1/5 is less than 1, the series converges.
To find the sum of a converging infinite geometric series, we can use the formula: S = a / (1 - r), where 'a' represents the first term and 'r' represents the common ratio.
In this series, the first term (a) is 150, and the common ratio (r) is 1/5. So, the sum (S) can be calculated as follows:
S = 150 / (1 - 1/5)
S = 150 / (4/5)
S = 150 * (5/4)
S = 750 / 4
S = 187.5
Therefore, the infinite geometric series 150 + 30 + 6 + ... converges to a sum of 187.5.