https://file.aiquickdraw.com/mathbot/file/1727625574653fmdyndjj.jpg

The problem provided involves an infinite geometric series. Let's break it down and solve it step by step.

### Given Series:
\[
\sum_{k=0}^{\infty} \left(\frac{-13}{5}\right)^k
\]

This is a geometric series with the general form:
\[
\sum_{k=0}^{\infty} ar^k = \frac{a}{1 - r}
\]
where \( a \) is the first term and \( r \) is the common ratio.

### Step 1: Identifying \( a \) and \( r \)
In this case:
- The first term \( a = 1 \) (because when \( k = 0 \), the term is \( 1 \)).
- The common ratio \( r = \frac{-13}{5} \).

### Step 2: Checking for Convergence
A geometric series converges if and only if the absolute value of the common ratio \( |r| < 1 \).

Here, the common ratio is \( \frac{-13}{5} \), and:
\[
\left|\frac{-13}{5}\right| = \frac{13}{5} = 2.6
\]

Since \( |r| = 2.6 > 1 \), **the series diverges**.

### Final Answer:
- The series **diverges**.
- Therefore, the sum of the series does not exist (DNE).

Would you like further clarification or details on this topic?

Here are some related questions to explore:
1. What are the conditions for a geometric series to converge?
2. How can you determine the sum of a converging geometric series?
3. What happens when the common ratio \( r \) is exactly 1?
4. Can a geometric series with a negative ratio converge?
5. How does the behavior of infinite series differ for different types of series?

**Tip:** For any geometric series, always check the absolute value of the common ratio to determine if the series converges or diverges.

Given the series: \( \sum_{k=0}^{\infty} \left( \frac{-13}{5} \right)^k \), does this series converge or diverge? If the series converges, find the sum of the series.

Explore the convergence and divergence of the infinite geometric series with a common ratio of -13/5. This problem demonstrates the conditions for a geometric series to diverge, using the formula for the sum of an infinite geometric series.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation