https://r2.erweima.ai/mathbot/image/172591589074975zrgi7z.jpg

The question is asking about infinite series and which statement is correct. Let's go through each of the given options one by one:

1. **"As séries infinitas precisam começar com índice 1."**
   - **Incorrect**. Infinite series do not necessarily need to start with index 1. They can start at any index, such as 0 or any positive integer.

2. **"Uma série diverge se \( S_n = S \)"**
   - **Incorrect**. If \( S_n = S \), this implies that the partial sums \( S_n \) approach a finite limit \( S \), meaning the series **converges**. A series diverges when the partial sums do not approach any finite limit.

3. **"A série \( \sum_{n=1}^{\infty} (-1)^{n-1} \) converge a 0."**
   - **Incorrect**. The given series is an alternating series of the form \( \sum (-1)^{n-1} \), which does not converge. It oscillates between 1 and -1 and does not settle to any value, so it does not converge to 0 or any other number.

4. **"A série geométrica sempre converge."**
   - **Incorrect**. A geometric series only converges if the common ratio \( |r| < 1 \). If \( |r| \geq 1 \), the series diverges.

5. **"A série \( \sum_{n=0}^{\infty} \left(\frac{2}{3}\right)^n \) converge."**
   - **Correct**. This is a geometric series with a common ratio \( r = \frac{2}{3} \), and since \( |r| < 1 \), the series converges.

### Conclusion:
The correct alternative is the last one: **"A série \( \sum_{n=0}^{\infty} \left(\frac{2}{3}\right)^n \) converge."**

Would you like further details or clarifications?

---

Here are 5 related questions to explore further:
1. What is the criterion for the convergence of geometric series?
2. How do we determine if an alternating series converges?
3. What is the difference between absolute and conditional convergence?
4. How does the partial sum \( S_n \) relate to the convergence of an infinite series?
5. What are some common tests to check the convergence of series?

**Tip**: To determine if a geometric series converges, always check the absolute value of the common ratio \( r \). If \( |r| < 1 \), the series converges.

A respeito das séries infinitas, assinale a alternativa correta.

This page provides a detailed analysis of infinite series, focusing on geometric series and the criteria for convergence or divergence. Learn how to identify whether a series converges based on the common ratio, and explore alternating series through mathematical examples.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation