We are tasked with determining the nature of the series whose general terms are provided in the image. Let's analyze each one carefully.

1. $u_n = \left( \frac{n}{n+1} \right)^{n^2}$

Analysis:

• Behavior of $\frac{n}{n+1}$: $\frac{n}{n+1} = 1 - \frac{1}{n+1}$ For large $n$, $\frac{n}{n+1} \to 1$, but we are raising it to the power $n^2$, so the decay rate is critical to analyze.

• Approximation for large $n$: Using $\ln(1-x) \approx -x$ for small $x$, we have: $\ln\left(\frac{n}{n+1}\right) = \ln\left(1 - \frac{1}{n+1}\right) \approx -\frac{1}{n+1}.$ Thus: $\left( \frac{n}{n+1} \right)^{n^2} = e^{n^2 \ln\left(\frac{n}{n+1}\right)} \approx e^{-n^2 \cdot \frac{1}{n+1}} \approx e^{-n}.$

• Nature of the series: The term $u_n \sim e^{-n}$ decays very quickly to 0. Since the series $\sum e^{-n}$ converges (it behaves like a geometric series with a ratio less than 1), the given series converges.

2. $u_n = \frac{1}{n \cos^2(n)}$

Analysis:

• Behavior of $\cos^2(n)$: The function $\cos^2(n)$ oscillates between 0 and 1, and it can get arbitrarily close to 0 as $n \to \infty$. This means $u_n$ can become arbitrarily large.

• Divergence test: Since $u_n$ does not tend to 0 due to the oscillatory nature of $\cos^2(n)$, the series diverges.

3. $u_n = \frac{1}{(\ln(n)) \ln(\ln(n))}$

Analysis:

• Growth of the denominator: For large $n$, $\ln(n) \to \infty$ and $\ln(\ln(n)) \to \infty$, so $u_n \to 0$. However, we must analyze the rate of decay to determine convergence.

• Comparison to a standard series: Compare $u_n$ with $\frac{1}{\ln(n)}$ or $\frac{1}{n^p}$:

  • The term $\frac{1}{\ln(n)}$ is known to diverge, and adding $\ln(\ln(n))$ in the denominator slows down the decay even more.
  • Thus, $u_n$ behaves similarly to a divergent series, and the given series diverges.

Conclusion:

1. Converges
2. Diverges
3. Diverges

Would you like a detailed explanation for any specific part?

Related Questions:

1. How can the comparison test be applied to identify convergence or divergence of a series?
2. Why does the oscillation in $\cos^2(n)$ cause divergence in the second series?
3. What is the role of logarithmic functions in slowing down series decay?
4. Can the ratio test be used for all three series? If so, how would it apply?
5. How does the exponential decay in the first series ensure convergence?

Tip:

When analyzing series with complex terms, approximating behavior for large $n$ using logarithms or Taylor expansions is often helpful to simplify the problem.