Which of the following statements is true for the series the summation from n=0 to infinity of (-1)^n and 5/4^n?
a) The series diverges because it is geometric with r = 5/4 and a = –1.
b) The series converges to 4 because it is geometric with r = -1/4 and a = 5.
c) The series converges to –1 because it is geometric with r = -1/4 and
a = -5/4.
d) The series diverges because it is geometric with r = -5/4 a = 1.
well, r = -5/4, so ...
Isn't the series convergent though?
To determine which statement is true for the series, we need to determine whether the series converges or diverges and find its sum if it converges.
The given series is the summation from n=0 to infinity of (-1)^n and 5/4^n.
First, let's check if the series is geometric by comparing consecutive terms. For a series to be geometric, the ratio between consecutive terms should be constant.
Let's calculate the ratio (r) between consecutive terms:
r = (5/4) / (-1) = -5/4.
Since the ratio is not constant, the series is not geometric.
Therefore, statement a) "The series diverges because it is geometric with r = 5/4 and a = –1" is not true.
Now, let's check the other statements.
Statement b) "The series converges to 4 because it is geometric with r = -1/4 and a = 5."
Since the series is not geometric, statement b) is not true.
Statement c) "The series converges to –1 because it is geometric with r = -1/4 and a = -5/4."
Again, the series is not geometric, so statement c) is not true.
Statement d) "The series diverges because it is geometric with r = -5/4 and a = 1."
Since the series is not geometric, statement d) is not true.
In conclusion, none of the given statements is true for the series.
To determine which statement is true for the given series, we need to analyze its convergence or divergence by considering its common ratio (r) and initial term (a).
The series can be written as the summation from n=0 to infinity of (-1)^n * (5/4)^n.
By looking at the form of the series, we can observe that it is a geometric series, where each term is a constant multiple of the previous term.
To determine the common ratio (r) of the series, we can take the ratio between any two consecutive terms. Let's consider the ratio of the (n+1)-th term to the n-th term:
((-1)^(n+1) * (5/4)^(n+1)) / ((-1)^n * (5/4)^n)
Simplifying this expression, we can cancel out (-1)^n:
((-1) * (5/4)^n * (5/4)) / ((5/4)^n)
= (-5/4) * (5/4)^n / (5/4)^n
= (-5/4)
Therefore, the common ratio (r) of the series is -5/4.
Now, let's analyze each statement with this information:
a) The statement claims that the series diverges because it is geometric with r = 5/4 and a = -1. This statement is incorrect because the correct common ratio is -5/4, not 5/4.
b) The statement claims that the series converges to 4 because it is geometric with r = -1/4 and a = 5. This statement is incorrect because the common ratio is -5/4, not -1/4.
c) The statement claims that the series converges to -1 because it is geometric with r = -1/4 and a = -5/4. This statement is incorrect because the common ratio is -5/4, not -1/4.
d) The statement claims that the series diverges because it is geometric with r = -5/4 and a = 1. This statement is correct because the common ratio is -5/4.
Therefore, the correct statement is (d) The series diverges because it is geometric with r = -5/4 and a = 1.