Consider the sequence \[ a_{n}=\frac{n \cdot \cos (n)}{5 n+10} \] Describe the behavior of the sequence. Is the sequence monotone? Monotone decreasing for all n Is the sequence bounded? Select Determine whether the sequence converges or diverges. If it converges, find the value it converges to. If it diverges, enter DIV.

Consider the sequence \[ a_n = \frac{n \cdot \cos(n)}{5n+10}. \] **Step 1. Rewrite the sequence.** Divide the numerator and the denominator by \(n\): \[ a_n = \frac{n \cdot \cos(n)}{5n+10} = \frac{\cos(n)}{5+\frac{10}{n}}. \] **Step 2. Behavior for large \(n\).** As \(n \to \infty\), the term \(\frac{10}{n} \to 0\). Thus, for large \(n\) the sequence behaves like: \[ a_n \approx \frac{\cos(n)}{5}. \] Since \(\cos(n)\) oscillates between \(-1\) and \(1\), the sequence oscillates between: \[ -\frac{1}{5} \quad \text{and} \quad \frac{1}{5}. \] **Step 3. Monotonicity.** The term \(\cos(n)\) does not behave in a monotonic manner; instead, it oscillates as \(n\) increases. Multiplying by constant factors or dividing by a term that approaches 5 does not remove the oscillations. Thus, the sequence is **not monotone** (it is not monotone decreasing for all \(n\)). **Step 4. Boundedness.** Since \[ -1 \leq \cos(n) \leq 1, \] we have \[ -\frac{1}{5+\frac{10}{n}} \leq a_n \leq \frac{1}{5+\frac{10}{n}}. \] For all \(n\), this gives a bound and as \(n\) increases, the bounds approach \(-\frac{1}{5}\) and \(\frac{1}{5}\). Hence, the sequence is **bounded**. **Step 5. Convergence.** A sequence converges if there exists a unique number \(L\) such that as \(n \to \infty\), \(a_n \to L\). In this case, because \(\cos(n)\) does not converge (it oscillates between \(-1\) and \(1\)), the fraction \(\frac{\cos(n)}{5+\frac{10}{n}}\) does not settle to a single value. Instead, it continues to oscillate between approximately \(-\frac{1}{5}\) and \(\frac{1}{5}\). Thus, the sequence **diverges**. **Summary:** - The sequence is **not monotone**. - The sequence is **bounded**. - The sequence **diverges** (does not converge to a single value).