Question

Determine whether the following sequence converges or diverges and describe whether it does do so monotonically or by oscillation. Give the limit when the sequence converges. {1. 000005^n) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. The sequence diverges by oscillation. The sequence converges monotonically. It converges to. The sequence converges by oscillation. It converges to square box. The sequence diverges monotonically.

Answer 1

The sequence {1.000005^n} diverges monotonically, because it increases without bound as n goes to infinity, and does not converge to a specific limit.

Step-by-step explanation:

To determine whether the following sequence converges or diverges and describe whether it does so monotonically or by oscillation, we analyze the given sequence {1.000005^n}. This sequence represents an exponential function with a base slightly greater than 1. As n increases, 1.000005^n also increases, but at an extremely slow rate since the base is very close to 1. Exponential functions of the form a^n with 0 < a < 1 decrease monotonically towards zero, while those with a > 1 increase monotonically towards infinity. Since the base here (1.000005) is greater than 1, the sequence is monotonic. It does not oscillate, as the term oscillation implies a sequence that repeatedly moves up and down between two values, which is not the case for exponential functions. Therefore, the given sequence diverges monotonically, because it increases without bound as n goes to infinity. This means the sequence does not converge to a specific finite limit.