CASE 2: $$ r

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CASE 2: $$ r<-1 $$ 1.3.1 If $$ r=-1,001 $$, determine the value of $$ r^{a} $$ if (a) $$ n=9999 $$ (b) $$ n=10000 $$ 1.3.2 Make a conjecture with regard to $$ r^{n} $$ and $$ S_{n} $$ as $$ n \rightarrow \infty $$

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**1.3.1 Evaluating $$ r^n $$ for $$ r = -1.001 $$:**

- **(a) For $$ n = 9999 $$:**  
  Since $$ 9999 $$ is odd,  
  $$
  (-1.001)^{9999} = -\,(1.001)^{9999}.
  $$
  That is, the result is negative with magnitude $$ (1.001)^{9999} $$.

- **(b) For $$ n = 10000 $$:**  
  Since $$ 10000 $$ is even,  
  $$
  (-1.001)^{10000} = (1.001)^{10000}.
  $$
  Here the result is positive with magnitude $$ (1.001)^{10000} $$.

*A useful observation:*  
Since  
$$
\ln(1.001) \approx 0.001,
$$
we have approximately
$$
(1.001)^{10000} \approx e^{0.001 \times 10000} = e^{10},
$$
and similarly  
$$
(1.001)^{9999} \approx e^{9.999}.
$$

---

**1.3.2 Conjecture on the behavior of $$ r^n $$ and $$ S_n $$ as $$ n \rightarrow \infty $$:**

- **For $$ r^n $$:**  
  When $$ r < -1 $$, the absolute value $$|r| > 1$$ implies that  
  $$
  |r^n| = |r|^n \rightarrow \infty \quad \text{as} \quad n \rightarrow \infty.
  $$
  Moreover, since $$ r $$ is negative, the sign of $$ r^n $$ alternates with $$ n $$ (negative when $$ n $$ is odd and positive when $$ n $$ is even). Thus, the sequence $$ \{r^n\} $$ oscillates between very large positive and negative values.

- **For $$ S_n $$:**  
  If we interpret $$ S_n $$ as the partial sum of the geometric series with first term $$ a $$ (commonly taken as $$ a=1 $$) and ratio $$ r $$,
  $$
  S_{n} = a \frac{1-r^{n}}{1-r},
  $$
  then the divergence of $$ r^n $$ (both in magnitude and oscillatory behavior) causes $$ S_{n} $$ to diverge as $$ n \rightarrow \infty $$. In other words, because $$ r^n $$ does not approach a finite limit, the series does not converge.

**Conjecture:**  
For $$ r < -1 $$, the individual term $$ r^n $$ does not tend to zero but rather oscillates in sign with increasing magnitude. Consequently, the geometric series $$ S_n $$ diverges as $$ n \rightarrow \infty $$.
**1.3.1 Evaluating $$ r^n $$ for $$ r = -1.001 $$:** - **(a) For $$ n = 9999 $$:** The result is negative with magnitude $$ (1.001)^{9999} $$. - **(b) For $$ n = 10000 $$:** The result is positive with magnitude $$ (1.001)^{10000} $$. **1.3.2 Conjecture on $$ r^n $$ and $$ S_n $$ as $$ n \rightarrow \infty $$:** - **For $$ r^n $$:** Since $$ r < -1 $$, $$ r^n $$ oscillates between very large positive and negative values as $$ n $$ increases. - **For $$ S_n $$:** The series $$ S_n $$ does not converge as $$ n $$ approaches infinity because $$ r^n $$ does not approach zero. **Conclusion:** For $$ r < -1 $$, $$ r^n $$ oscillates with increasing magnitude, and the geometric series $$ S_n $$ diverges.

CASE 2: \( r<-1 \) 1.3.1 If \( r=-1,001 \), determine the value of \( r^{a} \) if (a) \( n=9999 \) (b) \( n=10000 \) 1.3.2 Make a conjecture with regard to \( r^{n} \) and \( S_{n} \) as \( n \rightarrow \infty \)

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