Problem 27 - Precalculus Mathematics for Calculus

Question

If the sequence is convergent, find its limit. If it is divergent, explain why. $$a_{n}=\frac{1}{3^{n}}$$

Step-by-Step Solution

Answer

The sequence converges, and its limit is 0.

Step 1: Identify the Form of the Sequence

The given sequence is \( a_n = \frac{1}{3^n} \). It's a geometric sequence with the first term \(a_1 = \frac{1}{3}\) and a common ratio \(r = \frac{1}{3}\).

Step 2: Determine the Convergence of a Geometric Sequence

A geometric sequence \(a_n = a_1 \cdot r^{n-1}\) converges if the absolute value of the common ratio \( |r| < 1 \). In this sequence, \(|r| = \left|\frac{1}{3}\right| = \frac{1}{3} < 1\), thus the sequence converges.

Step 3: Find the Limit of the Convergent Sequence

For a geometric sequence with \(|r| < 1\), the limit as \(n\) approaches infinity is zero. Therefore, the limit of \(a_n = \frac{1}{3^n}\) is \( \lim_{{n \to \infty}} a_n = 0 \).

Key Concepts

Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant, known as the common ratio. An example looks like this: if the first term is 2 and the common ratio is 3, the sequence would be 2, 6, 18, 54, and so on.
In the given problem, the sequence is defined by \( a_n = \frac{1}{3^n} \). This sequence is geometric because each term is obtained by multiplying the previous term by \( \frac{1}{3} \). This specific sequence starts with \( a_1 = \frac{1}{3} \), and its common ratio \( r = \frac{1}{3} \).
Things to remember about geometric sequences:

• Every term is multiplied by the common ratio \( r \) to get the next term.
• The formula used is \( a_n = a_1 \cdot r^{n-1} \).
• If the common ratio is a fraction or a decimal between -1 and 1, various interesting properties arise, such as convergence.

Limit of a Sequence

The limit of a sequence is the value that the terms of the sequence approach as the sequence progresses towards infinity. It is a fundamental concept in calculus and helps us understand the behavior of sequences over the long term.
For the sequence defined by \( a_n = \frac{1}{3^n} \), we are concerned with what happens as \( n \) becomes very large. Each term gets increasingly smaller, hinting that the sequence could have a finite limit.
When finding the limit of a geometric sequence where the absolute value of the common ratio \( |r| < 1 \), the sequence approaches 0. So, for our sequence:

• The terms become negligible as \( n \) increases.
• The limit as \( n \to \infty \) is \( 0 \), showing the sequence's gradual tendency towards this value.

Convergent Sequence

A convergent sequence is one whose terms approach a specific value, known as the limit, as they progress to infinity. If a sequence does not approach a specific value, it is called divergent.
For instance, the sequence \( a_n = \frac{1}{3^n} \) is convergent because its terms approach 0 as \( n \) becomes larger without bound. This behavior is typical when the absolute value of the common ratio \( |r| < 1 \) for a geometric sequence, as mentioned previously.
Key aspects of convergent sequences:

• The sequence stabilizes around a particular value.
• Geometric sequences with a small common ratio (i.e., less than 1) tend to converge.
• Convergent sequences are predictable and behave nicely mathematically, allowing for easier analysis and calculation of their limits.