1.3.2 Make a conjecture with regard to $$ r^{n} $$ and $$ S_{n} $$ as $$ n \rightarrow \infty $$ (2) 1.4 CASE 3: $$ r=1 $$ 1.4.1 What is the sum of the geometric series $$ S_{n}=a+a r+a r^{2}+\ldots a r^{n-1} \text { if } r=1 ? $$ 1.4.2 Make a conjecture with regard to $$ r^{n} $$ and $$ S_{n} $$ as $$ n \rightarrow \infty $$ 1.5 CASE 4: $$ r=-1 $$ 1.5.1 What is the sum of the geometric series $$ S_{n}=a+a r+a r^{2}+\ldots a r^{n-1} \text { if } r=-1 ? $$ 1.5.2 Do the sums above approach some finite particular number as $$ n \rightarrow \infty $$ i.e. is the sequence divergent or convergent? 1.6 CASE 5: $$ -1

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### 1.3.2 Conjecture Regarding $$ r^{n} $$ and $$ S_{n} $$ as $$ n \rightarrow \infty $$

For a geometric series defined as:
$$
S_{n} = a + ar + ar^{2} + \ldots + ar^{n-1}
$$
where $$ a $$ is the first term and $$ r $$ is the common ratio, we can analyze the behavior of $$ r^{n} $$ and $$ S_{n} $$ as $$ n $$ approaches infinity.

- If $$ |r| < 1 $$: As $$ n \rightarrow \infty $$, $$ r^{n} \rightarrow 0 $$. Thus, $$ S_{n} $$ converges to $$ \frac{a}{1 - r} $$.
- If $$ |r| > 1 $$: As $$ n \rightarrow \infty $$, $$ r^{n} \rightarrow \infty $$. Thus, $$ S_{n} $$ diverges.
- If $$ r = 1 $$: $$ S_{n} = na $$ which diverges as $$ n \rightarrow \infty $$.
- If $$ r = -1 $$: The series oscillates between $$ 0 $$ and $$ a $$ depending on whether $$ n $$ is even or odd, thus it does not converge.

### 1.4 CASE 3: $$ r=1 $$

#### 1.4.1 Sum of the Geometric Series

If $$ r = 1 $$:
$$
S_{n} = a + a + a + \ldots + a = na
$$
Thus, the sum of the geometric series is $$ S_{n} = na $$.

#### 1.4.2 Conjecture Regarding $$ r^{n} $$ and $$ S_{n} $$ as $$ n \rightarrow \infty $$

As $$ n \rightarrow \infty $$:
- $$ r^{n} = 1^{n} = 1 $$
- $$ S_{n} = na \rightarrow \infty $$ (if $$ a \neq 0 $$)

### 1.5 CASE 4: $$ r=-1 $$

#### 1.5.1 Sum of the Geometric Series

If $$ r = -1 $$:
$$
S_{n} = a - a + a - a + \ldots
$$
This series alternates between $$ a $$ and $$ 0 $$ depending on whether $$ n $$ is even or odd. Therefore:
- If $$ n $$ is even, $$ S_{n} = 0 $$
- If $$ n $$ is odd, $$ S_{n} = a $$

#### 1.5.2 Convergence or Divergence

As $$ n \rightarrow \infty $$:
- The series does not converge to a single value; it oscillates between $$ 0 $$ and $$ a $$. Thus, it is divergent.

### 1.6 CASE 5: $$ -1**1.3.2 Conjecture:** As $$ n $$ becomes very large, $$ r^{n} $$ approaches 0 if $$ |r| < 1 $$, and the sum $$ S_{n} $$ approaches $$ \frac{a}{1 - r} $$. If $$ |r| \geq 1 $$, $$ S_{n} $$ does not approach a finite number. **1.4 CASE 3: $$ r=1 $$** - **Sum:** $$ S_{n} = na $$ - **Conclusion:** As $$ n $$ increases, $$ S_{n} $$ grows without bound. **1.5 CASE 4: $$ r=-1 $$** - **Sum:** Alternates between $$ a $$ and 0. - **Conclusion:** The series does not converge as $$ n $$ increases. **1.6 CASE 5: $$ -1 < r < 1 $$** - **Possible Values of $$ r $$:** 0.5, -0.5, 0.9 - **Paper Cutting Steps:** 1. Cut A4 into two $$ 8 \, \text{unit}^2 $$ rectangles. 2. Cut one into two $$ 4 \, \text{unit}^2 $$ rectangles. 3. Cut one into two $$ 2 \, \text{unit}^2 $$ rectangles. 4. Cut one into two $$ 1 \, \text{unit}^2 $$ rectangles. 5. Next three areas: $$ 1 \, \text{unit}^2 $$, $$ 0.5 \, \text{unit}^2 $$, $$ 0.25 \, \text{unit}^2 $$ - **Geometric Sequence Explanation:** Each area is half of the previous, forming a geometric sequence with a common ratio of $$ \frac{1}{2} $$.

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