The image you uploaded contains a question about the convergence of several series. The question presents four series, and the task is to determine which of the given alternatives is correct based on their convergence or divergence.

Let’s analyze each series:

Series I: $\sum_{n=1}^{\infty} \frac{5}{2^n}$

This is a geometric series with a ratio $r = \frac{1}{2}$. A geometric series converges if $|r| < 1$, so this series converges.

Series II: $\sum_{n=1}^{\infty} \frac{1}{2^n - 3^n}$

For large values of $n$, the dominant term in the denominator is $3^n$, so the series behaves like $\sum_{n=1}^{\infty} \frac{1}{-3^n}$, which is a geometric series with ratio $r = \frac{1}{3}$. Since $|r| < 1$, this series converges.

Series III: $\sum_{n=1}^{\infty} \frac{1}{2^n + 3^n}$

For large $n$, the dominant term in the denominator is $3^n$, so the series behaves like $\sum_{n=1}^{\infty} \frac{1}{3^n}$, which is also a geometric series with ratio $r = \frac{1}{3}$. This series converges.

Series IV: $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{\sqrt{n}}$

This is an alternating series. To determine whether it converges, we apply the alternating series test, which requires that the terms $\frac{1}{\sqrt{n}}$ decrease monotonically and approach zero as $n \to \infty$. Since both conditions are satisfied, this series converges.

Conclusion:

All four series converge, so the correct alternative is E (As séries (I), (II), e (IV) convergem).

Would you like further details or explanations on any of the series? Here are some related questions:

1. How does the alternating series test work?
2. What is the condition for convergence of a geometric series?
3. How do you find the limit of a series with a dominant term?
4. What are the characteristics of a series that diverges?
5. Can you compare the convergence of different types of series (e.g., geometric vs. alternating)?

Tip: For geometric series, always check the ratio $r$. If $|r| < 1$, the series converges; if $|r| \geq 1$, the series diverges.