https://file.aiquickdraw.com/mathbot/file/1727625496800s805w5i3.jpg

The series given in the image is:

\[
\sum_{k=0}^{\infty} 7 \left( \frac{4}{9} \right)^k
\]

This is a geometric series of the form:

\[
\sum_{k=0}^{\infty} ar^k
\]

Where:
- \( a = 7 \) (the first term of the series)
- \( r = \frac{4}{9} \) (the common ratio)

A geometric series converges if the absolute value of the common ratio \( |r| \) is less than 1. In this case:

\[
|r| = \left|\frac{4}{9}\right| = \frac{4}{9} < 1
\]

Since \( |r| < 1 \), the series converges. The sum of an infinite geometric series is given by the formula:

\[
S = \frac{a}{1 - r}
\]

Substituting the values of \( a \) and \( r \):

\[
S = \frac{7}{1 - \frac{4}{9}} = \frac{7}{\frac{9 - 4}{9}} = \frac{7}{\frac{5}{9}} = 7 \times \frac{9}{5} = \frac{63}{5}
\]

Thus, the sum of the series is:

\[
S = \frac{63}{5} = 12.6
\]

Let me know if you need further details!

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Here are 5 related questions:
1. How do you determine if a geometric series converges?
2. What happens to the sum formula if \( |r| \geq 1 \)?
3. Can a series with a negative common ratio converge?
4. How would you generalize the sum formula for any geometric series?
5. What is the sum of the series \( \sum_{k=0}^{\infty} 7 \left(\frac{2}{3}\right)^k \)?

**Tip:** Always check the value of the common ratio to decide if a geometric series converges or diverges.

The series ∑ (7 * (4/9)^k) from k = 0 to infinity. If the series converges, what is its sum?

Learn how to determine if an infinite geometric series converges and calculate its sum. This problem explores the series ∑ (7 * (4/9)^k) from k = 0 to infinity. Concepts include the formula for the sum of an infinite geometric series and the conditions for convergence.

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