Which of the following statements is true for the series the summation from n equals 0 to infinity of the product of negative 1 raised to the nth power and 3 over 2 to the nth power?
A. The series diverges because it is geometric with r = 3 over 2 and a = –1.
B. The series converges to –1 because it is geometric with r = negative 1 over 2 and a = negative 3 over 2.
C. The series diverges because it is geometric with r = negative 3 over 2 and a = 1.
D. The series converges to 2 because it is geometric with r = negative 1 over 2 and a = 3.
To determine whether the given series converges or diverges and find its sum, we need to analyze its terms.
The given series is defined as the summation from n equals 0 to infinity of (-1)^n * (3/2)^n.
Since the series involves the product of two terms (-1)^n and (3/2)^n, we can determine its convergence or divergence by analyzing the second term, (3/2)^n.
For a geometric series of the form sum of a*r^n, where a is the first term and r is the common ratio, the series converges if the absolute value of r is less than 1 and diverges if the absolute value of r is greater than or equal to 1.
In this case, the common ratio (3/2)^n approaches infinity as n approaches infinity, so its absolute value is greater than 1. Therefore, the given series diverges.
Therefore, the correct answer is:
C. The series diverges because it is geometric with r = -3/2 and a = 1.