https://file.aiquickdraw.com/mathbot/file/1726414022199e1ce0brf.jpg

kerjakan tiap soal tersebut untuk membuktikan bahwa tiap soal itu divergen atau konvergen.

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.

Kerjakan tiap soal tersebut untuk membuktikan bahwa tiap soal itu divergen atau konvergen.

Explore the convergence and divergence of different series, including geometric series, harmonic series, and telescoping series. This exercise helps understand series behavior using formulas and theorems. Suitable for undergraduate math students.

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