I will analyze the series in the image one by one to determine if they are convergent or divergent.

1. Series 1: $\sum_{n=1}^{\infty} 2^{2n} \cdot 3^{1-n}$ We can rewrite the general term: $a_n = 2^{2n} \cdot 3^{1-n} = \frac{4^n}{3^{n-1}} = 4^n \cdot 3^{-n+1}$ This behaves like an exponential series where the base is greater than 1, which suggests it diverges.

2. Series 2: $\sum_{n=0}^{\infty} x^n$ This is a geometric series with a common ratio $x$. For a geometric series to converge, $|x| < 1$. So:

   • If $|x| < 1$, the series converges.
   • If $|x| \geq 1$, the series diverges.

3. Series 3: $\sum_{n=1}^{\infty} \frac{1}{n(n+1)}$ We can simplify the general term using partial fractions: $\frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1}$ This is a telescoping series, where most terms cancel out. The series converges because the remaining terms approach zero.

4. Series 4: $\sum_{n=1}^{\infty} \frac{1}{n}$ This is the harmonic series, which is known to diverge.

5. Series 5: $\sum_{n=1}^{\infty} \left(\frac{3}{n(n+1)} + \frac{1}{2^n}\right)$ The first part of the sum is again a telescoping series, which converges. The second part, $\sum \frac{1}{2^n}$, is a geometric series with a ratio $r = \frac{1}{2}$, so it also converges. Therefore, this series converges.

Summary:

1. Divergent.
2. Converges if $|x| < 1$, diverges otherwise.
3. Convergent.
4. Divergent.
5. Convergent.

Would you like any additional details or clarifications?

Here are 5 related questions you could explore:

1. What is the test for convergence of a geometric series?
2. How can we determine the convergence of a series using the ratio test?
3. What is the importance of partial fractions in simplifying series?
4. What are the conditions for divergence in a p-series?
5. How do telescoping series simplify convergence proofs?

Tip: When dealing with infinite series, checking the form of the general term can give you insights into whether the series converges or diverges.