To analyze the convergence of the series

$\sum_{n=1}^\infty \frac{\cos(1)}{n},$

we note that the term $\cos(1)$ is a constant, independent of $n$. Hence, the series can be written as:

$\cos(1) \sum_{n=1}^\infty \frac{1}{n}.$

The convergence of the series depends on the behavior of $\sum_{n=1}^\infty \frac{1}{n}$, which is the harmonic series.

Convergence Analysis:

The harmonic series $\sum_{n=1}^\infty \frac{1}{n}$ diverges. This means that as $n \to \infty$, the partial sums of the series grow without bound. Since multiplying a divergent series by a constant $\cos(1)$ does not change its divergence, the given series also diverges.

Conclusion:

The series $\sum_{n=1}^\infty \frac{\cos(1)}{n}$ diverges.

Would you like further explanation or details about series convergence?

Here are five related questions to explore:

1. What is the harmonic series, and why does it diverge?
2. How does the comparison test apply to determine series convergence or divergence?
3. What happens if we replace $\cos(1)$ with a variable term depending on $n$?
4. Can we determine if $\sum_{n=1}^\infty \frac{\cos(1)}{n}$ converges absolutely or conditionally?
5. What role do p-series play in evaluating convergence?

Tip: A good rule to remember is that the harmonic series $\sum_{n=1}^\infty \frac{1}{n}$ diverges, even though its terms tend to zero.