https://file.aiquickdraw.com/mathbot/file/17262525081459be1y1qi.jpg

Let's break down the given geometric series and address the two parts of the question:

### Given Geometric Series:
The series is presented as:
\[
2 - (x+1) + \frac{(x+1)^2}{2} - \frac{(x+1)^3}{2^2} + \cdots + \frac{[-(x+1)]^{r-1}}{2^{r-2}} + \cdots
\]
This is a geometric series where the first term \( a = 2 \) and the common ratio \( r \) can be identified by examining the successive terms.

### (i) Find the set of values of \( x \) such that the sum to infinity of the series exists (i.e., the series converges):

For a geometric series to converge, the absolute value of the common ratio \( |r| \) must be less than 1:
\[
|r| < 1
\]

From the given series, the common ratio is:
\[
r = -\frac{x+1}{2}
\]
For convergence, we require:
\[
\left| -\frac{x+1}{2} \right| < 1
\]
This simplifies to:
\[
\frac{|x+1|}{2} < 1
\]
Multiplying both sides by 2:
\[
|x+1| < 2
\]
This inequality splits into two parts:
\[
-2 < x+1 < 2
\]
Subtracting 1 from all sides:
\[
-3 < x < 1
\]
Thus, the series converges for \( x \) in the interval:
\[
\boxed{-3 < x < 1}
\]

### (ii) Find the value of \( x \) for which the sum to infinity \( S_{\infty} = \frac{4}{3} \):

The sum to infinity of a geometric series is given by the formula:
\[
S_{\infty} = \frac{a}{1 - r}
\]
We are given that \( S_{\infty} = \frac{4}{3} \), the first term \( a = 2 \), and the common ratio \( r = -\frac{x+1}{2} \).

Substitute into the sum formula:
\[
\frac{4}{3} = \frac{2}{1 - \left( -\frac{x+1}{2} \right)}
\]
Simplify:
\[
\frac{4}{3} = \frac{2}{1 + \frac{x+1}{2}}
\]
Multiply both sides by \( 1 + \frac{x+1}{2} \):
\[
\frac{4}{3} \left( 1 + \frac{x+1}{2} \right) = 2
\]
Distribute \( \frac{4}{3} \):
\[
\frac{4}{3} + \frac{4(x+1)}{6} = 2
\]
Multiply the entire equation by 6 to eliminate the denominators:
\[
8 + 4(x+1) = 12
\]
Simplify:
\[
4(x+1) = 4
\]
Divide by 4:
\[
x+1 = 1
\]
Thus:
\[
x = 0
\]

The value of \( x \) for which the sum to infinity is \( \frac{4}{3} \) is:
\[
\boxed{0}
\]

---

Would you like further explanation on any of the steps? Here are five related questions to expand on this topic:

1. How is the formula for the sum to infinity of a geometric series derived?
2. Why does the common ratio of a geometric series need to satisfy \( |r| < 1 \) for convergence?
3. Can you solve a similar series if the first term is not a constant but depends on \( x \)?
4. What happens to the series when \( x \) equals -3 or 1, where the convergence condition fails?
5. How does changing the common ratio affect the sum to infinity of a geometric series?

**Tip:** Always check if a geometric series converges before attempting to calculate the sum to infinity!

A geometric series is given as follows: 2 - (x + 1) + (x + 1)^2 / 2 - (x + 1)^3 / 2^2 + ... Find the set of values of x such that the sum to infinity of the series exists (i.e., the series converges) and find the value of x for which the sum to infinity is 4/3.

Solve a geometric series problem to find the range of x for which the series converges and the specific value of x that results in a sum to infinity of 4/3. The problem explores series convergence, sum to infinity formulas, and algebraic manipulation.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation