Determine whether the following sequence converges or diverges, and state whether it is monotonic or whether it oscillates. Give the limit if the sequence converges. \[ \left\{(-1.000003)^{n}\right\} \] Select the correct choice and, if necessary, fill in the answer box to complete your choice. A. The sequence converges to \( \square \) is monotonic, and does not oscillate. B. The sequence diverges, is monotonic, and does not oscillate. C. The sequence diverges, is monotonic, and oscillates. D. The sequence diverges, is not monotonic, and does not oscillate. E. The sequence converges to \( \square \) , is not monotonic, and oscillates. F. The sequence converges to \( \square \) , is monotonic, and oscillates. G. The sequence diverges, is not monotonic, and oscillates, H. The sequence converges to \( \square \) , is not monotonic, and does not oscillate.
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To analyze the sequence \((-1.000003)^{n}\), we can see that as \(n\) takes integer values, the term oscillates between two values: for even \(n\), it is close to 1 (since \((-1.000003)^{2k} \approx 1.000006\)), and for odd \(n\), it approaches -1 (since \((-1.000003)^{2k+1} \approx -1.000003\)). Therefore, the sequence does not converge to a single limit, and it oscillates between these two values, hence it diverges. Additionally, oscillating sequences are not monotonic since they go back and forth between their limits rather than increasing or decreasing consistently. So, the correct choice is: G. The sequence diverges, is not monotonic, and oscillates.
