https://r2.erweima.ai/mathbot/image/1727195059244yglxaiiz.jpg

The image contains two distinct problems:

1. **The first expression**:
   \[
   5(-5) + 5(-5)^2 + 5(-5)^3 + \dots
   \]
   This appears to be an infinite series with alternating powers of \(-5\). The terms are in the form of \( 5(-5)^n \), which could suggest a geometric series.

2. **The second expression**:
   \[
   \sum_{j=1}^{\infty} 3\left( \frac{-2}{3} \right)^j
   \]
   This is also an infinite series, where the general term is \( 3 \left( \frac{-2}{3} \right)^j \), a geometric series.

Let's start by analyzing and solving each one. I will compute the sums for both series and check whether they converge.### Solutions:
1. **First Series**: 
   \[
   5(-5) + 5(-5)^2 + 5(-5)^3 + \dots
   \]
   This series does **not converge** because the common ratio is \(-5\), which has an absolute value greater than 1. Therefore, the sum does not exist.

2. **Second Series**: 
   \[
   \sum_{j=1}^{\infty} 3\left( \frac{-2}{3} \right)^j
   \]
   This is a geometric series with a first term of \(-2\) and a common ratio of \(-\frac{2}{3}\), which satisfies the condition for convergence \(|r| < 1\). The sum of the series is:
   \[
   -1.2
   \]

Would you like further details or clarification on these calculations?

Here are 5 related questions to consider:
1. What conditions are necessary for a geometric series to converge?
2. How can we determine the divergence of a series?
3. Can we extend the concept of geometric series to complex numbers?
4. How does the value of the common ratio affect the behavior of a series?
5. In what situations might an infinite series not have a sum?

**Tip**: Always check the common ratio of a geometric series to ensure it meets the convergence criteria \( |r| < 1 \).

5(-5) + 5(-5)^2 + 5(-5)^3 + ... and ∑(j=1 to ∞) 3(-2/3)^j

Solve two infinite series problems: one with powers of -5 and another with a ratio of -2/3. Learn about the conditions for geometric series convergence and calculate exact sums where applicable.

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