https://file.aiquickdraw.com/mathbot/file/1727514078694joek7mnm.jpg

Let's analyze the infinite geometric series given in the image:

\[
1 + \left(\frac{2x}{3}\right) + \left(\frac{2x}{3}\right)^2 + \left(\frac{2x}{3}\right)^3 + \dots
\]

### (a) For what values of \( x \) does the series converge?

This is a geometric series of the form:

\[
S = 1 + r + r^2 + r^3 + \dots
\]

where \( r = \frac{2x}{3} \). A geometric series converges if the absolute value of the common ratio \( r \) is less than 1:

\[
|r| < 1
\]

Substituting \( r = \frac{2x}{3} \):

\[
\left| \frac{2x}{3} \right| < 1
\]

To find the range of \( x \), we solve:

\[
\frac{|2x|}{3} < 1
\]

Multiplying both sides by 3:

\[
|2x| < 3
\]

Dividing by 2:

\[
|x| < \frac{3}{2}
\]

Thus, the series converges when:

\[
- \frac{3}{2} < x < \frac{3}{2}
\]

### (b) Find the sum of the series if \( x = 1.2 \).

The sum \( S \) of an infinite geometric series is given by the formula:

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

where \( a \) is the first term (which is 1 in this case) and \( r \) is the common ratio. Here, \( r = \frac{2x}{3} \), so for \( x = 1.2 \):

\[
r = \frac{2(1.2)}{3} = 0.8
\]

Now, applying the formula:

\[
S = \frac{1}{1 - 0.8} = \frac{1}{0.2} = 5
\]

Thus, the sum of the series when \( x = 1.2 \) is \( S = 5 \).

---

Would you like more details on any of these steps? Here are 5 related questions to explore this concept further:

1. How does the common ratio affect the convergence of a geometric series?
2. What happens to the sum if \( x \) approaches \( \frac{3}{2} \)?
3. Can you explain the derivation of the sum formula for a geometric series?
4. How would the series change if the first term wasn't 1?
5. What would be the sum if \( x = -1.4 \)?

**Tip:** When dealing with geometric series, always verify the condition for convergence before attempting to find the sum.

Consider the infinite geometric series 1 + (2x/3) + (2x/3)^2 + (2x/3)^3 + ..... (a) For what values of x does the series converge? (b) Find the sum of the series if x = 1.2.

Explore how to determine the values of x for which an infinite geometric series converges and calculate the sum when x = 1.2. Learn key concepts related to geometric series, including the convergence condition and the sum formula.

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