Question

Which of the following statements is true concerning the series ∑_{n=10}^{[infinity]} ((-1)^{n}+10) / 5^n ? (A) The series converges by the Limit Comparison Test when comparing with b

Answer 1

The given series ∑_{n=10}^{[infinity]} ((-1)^{n}+10) / 5^n converges by the Limit Comparison Test when comparing with the series ∑_{n=10}^{[infinity]} 1 / 5^n.

Step-by-step explanation:

The given series is ∑_{n=10}^{[infinity]} ((-1)^{n}+10) / 5^n.

We can use the limit comparison test to determine if the series converges or diverges. Let's compare the given series with the series b = ∑_{n=10}^{[infinity]} 1 / 5^n.

By taking the limit as n approaches infinity, we find that the limit of ((-1)^{n}+10) / 5^n divided by 1 / 5^n is 1.

Since this limit is a finite positive number, the given series converges if the series b converges. The series b is a geometric series with a common ratio of 1/5, which converges. Therefore, the statement is true.