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 $$ L=\square $$.

Question

UpStudy AI · Accepted Answer

We are given the sequence

$$
a_m = (-0.006)^m.
$$

**Step 1. Consider the Absolute Value**

The absolute value of the sequence term is

$$
|a_m| = |(-0.006)^m| = (0.006)^m.
$$

Since $$0.006 < 1$$, we have

$$
\lim_{m\to\infty}(0.006)^m = 0.
$$

**Step 2. Determine the Sign Behavior**

Because the base $$-0.006$$ is negative, the sign of each term alternates with $$m$$. For even $$m$$,

$$
(-0.006)^m > 0,
$$

and for odd $$m$$,

$$
(-0.006)^m < 0.
$$

Thus, even though the magnitude tends to zero, the sequence does not approach zero in a monotonic (only decreasing or increasing) manner. Instead, it oscillates between positive and negative values.

**Step 3. Convergence Analysis**

Since the magnitude of the sequence tends to zero and the oscillation decreases as $$m$$ increases (because the absolute values become smaller and approach zero), the sequence converges to

$$
L = 0.
$$

However, it converges by oscillation rather than monotonically.

**Final Choice**

The sequence converges by oscillation to the limit $$L=0$$.

Thus, the correct choice is:

D. The sequence $$ \left\{(-0.006)^{m}\right\} $$ converges by oscillation to the limit $$ L = 0 $$.
The sequence $$ \left\{(-0.006)^{m}\right\} $$ converges by oscillation to the limit $$ L = 0 $$.

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