To solve these problems, we need to evaluate the limits of each given geometric sequence as $n \to \infty$. Here’s the analysis for each:

Problem (1)

$\lim_{{n \to \infty}} \left( \frac{1}{3} \right)^n$ Since $\frac{1}{3} < 1$, as $n$ approaches infinity, $\left( \frac{1}{3} \right)^n$ will approach 0. Therefore: $\lim_{{n \to \infty}} \left( \frac{1}{3} \right)^n = 0$

Problem (2)

$\lim_{{n \to \infty}} 3^n$ In this case, $3 > 1$, so as $n$ approaches infinity, $3^n$ will grow without bound and approach infinity. Thus: $\lim_{{n \to \infty}} 3^n = \infty$

Problem (3)

$\lim_{{n \to \infty}} \left( \frac{2}{3} \right)^n$ Here, $\frac{2}{3} < 1$, and as $n$ approaches infinity, $\left( \frac{2}{3} \right)^n$ will approach 0. Therefore: $\lim_{{n \to \infty}} \left( \frac{2}{3} \right)^n = 0$

Problem (4)

$\lim_{{n \to \infty}} \left( \frac{3}{2} \right)^n$ Since $\frac{3}{2} > 1$, $\left( \frac{3}{2} \right)^n$ will grow without bound as $n$ approaches infinity, tending toward infinity. Thus: $\lim_{{n \to \infty}} \left( \frac{3}{2} \right)^n = \infty$

Summary of Results:

1. $\lim_{{n \to \infty}} \left( \frac{1}{3} \right)^n = 0$
2. $\lim_{{n \to \infty}} 3^n = \infty$
3. $\lim_{{n \to \infty}} \left( \frac{2}{3} \right)^n = 0$
4. $\lim_{{n \to \infty}} \left( \frac{3}{2} \right)^n = \infty$

Would you like a detailed explanation of each step, or do you have any questions?

Related Questions:

1. What would happen if the base of each sequence were exactly equal to 1?
2. How would the limits change if the base were a negative number between -1 and 1?
3. Can you find a general rule for when a geometric sequence tends to zero, infinity, or remains constant?
4. How does the rate at which a sequence approaches infinity differ for different values greater than 1?
5. How would adding a constant to each sequence term (e.g., $(\frac{1}{3})^n + 5$) affect the limit?

Tip:

For any geometric sequence $a^n$ as $n \to \infty$, if $|a| < 1$, the limit is 0; if $a = 1$, the limit is 1; and if $|a| > 1$, the limit tends to infinity (or diverges if $a$ is negative).