Determine whether the sequence \( \left\{(-0.006)^{m}\right\} \) converges or diverges and describe whether it does so monotonically or by oscillation. Give the limit if the sequence converges. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The sequence \( \left\{(-0.006)^{m}\right\} \) converges monotonically to the limit \( L= \) B. The sequence \( \left\{(-0.006)^{m}\right\} \) diverges monotonically. C. The sequence \( \left\{(-0.006)^{m}\right\} \) diverges by oscillation. D. The sequence \( \left\{(-0.006)^{m}\right\} \) converges by oscillation to the limit \( L=\square \).
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To determine the behavior of the sequence \( \left\{(-0.006)^{m}\right\} \), we note that since \( -0.006 < 0 \), the terms will alternate in sign as \( m \) increases. For even \( m \), the terms are positive, and for odd \( m \), they are negative. As \( m \to \infty \), the magnitude \( |(-0.006)^{m}| = 0.006^m \) approaches zero due to the base being less than 1. Therefore, this sequence converges, but due to the alternating signs, it converges by oscillation. The correct choice is: D. The sequence \( \left\{(-0.006)^{m}\right\} \) converges by oscillation to the limit \( L= 0 \).
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