Problem 31 - Calculus: Early Transcendentals

Question

Determine whether the sequence converges or diverges. If it converges, find the limit. \( a_n = \frac {4^n}{1 + 9^n} \)

Step-by-Step Solution

Answer

The sequence converges to 0.

Step 1: Identify the Sequence Type

The sequence is given by \( a_n = \frac{4^n}{1 + 9^n} \). Notice that the numerator and the denominator are both exponential functions of \( n \). This suggests we analyze the growth rates of \( 4^n \) and \( 9^n \).

Step 2: Compare Growth Rates

Both \( 4^n \) and \( 9^n \) are exponential functions, but \( 9^n \) grows more quickly than \( 4^n \) because the base of 9 is larger than 4. As \( n \rightarrow \infty \), \( 9^n \) dominates \( 4^n \) in the expression.

Step 3: Apply Limit Analysis

Consider the limit \( \lim_{n \to \infty} \frac{4^n}{1 + 9^n} \). As \( 9^n \) dominates, \( 1 + 9^n \) behaves like \( 9^n \) for large \( n \). Therefore, the limit simplifies to \( \lim_{n \to \infty} \frac{4^n}{9^n} = \lim_{n \to \infty} \left( \frac{4}{9} \right)^n \).

Step 4: Evaluate the Simplified Limit

The term \( \left( \frac{4}{9} \right)^n \) is a decreasing exponential sequence, because \( \frac{4}{9} < 1 \). As \( n \rightarrow \infty \), \( \left( \frac{4}{9} \right)^n \rightarrow 0 \). Thus, the limit of the sequence is 0.

Step 5: State the Conclusion

The sequence \( a_n = \frac{4^n}{1 + 9^n} \) converges as \( n \rightarrow \infty \), and its limit is 0.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to the power of a variable exponent. In our sequence example, both the numerator and the denominator are in exponential form:

• The numerator is given by the exponential function \(4^n\), where 4 is the base and \(n\) is the exponent.
• The denominator includes the term \(9^n\), with 9 as a base and \(n\) as the exponent.

An interesting property of exponential functions is their growth behavior. For positive bases greater than one, these functions increase rapidly as the exponent grows larger.
This rapid growth makes exponential functions pivotal in scenarios involving repeated multiplication, such as compound interest calculations and population growth models.
Understanding exponential functions involves recognizing how different bases affect their growth rates over time, which brings us to our next important concept.

Growth Rates Comparison

Comparing growth rates helps us understand how different exponential functions behave relative to one another.
In the sequence given, comparing the exponential terms \(4^n\) and \(9^n\) is crucial:

• The base of \(9^n\) is larger than that of \(4^n\), which implies that \(9^n\) increases faster than \(4^n\) as \(n\) becomes quite large.

Growth rates become especially significant when determining the behavior of a sequence as its variable approaches infinity.
In practical terms:

• As time goes on (or \(n\) increases), the term with the larger base (here \(9^n\)) will come to dominate the expression \(1 + 9^n\) as if the \(1\) is negligible by comparison.

Grasping growth rates of exponential functions allows for the evaluation of sequence convergence, which is closely tied to assessing the eventual behavior of a sequence.

Limit Evaluation

Limit evaluation deals with determining the behavior of a sequence or function as its variable approaches a certain point, often infinity.
In this exercise, the limit of the sequence \(a_n = \frac{4^n}{1 + 9^n}\) as \(n\) approaches infinity is what we need to discover.

• Simplifying the limit involves recognizing that \(9^n\) will overwhelmingly dominate the term \(1 + 9^n\), effectively making it behave like \(9^n\).
• This simplifies the limit calculation to \(\lim_{n \to \infty} \frac{4^n}{9^n}\), which reduces to \(\left(\frac{4}{9}\right)^n\). This term represents a fraction raised to an exponential power.

As \(\frac{4}{9}\) is a number less than one, raising it to an increasingly large power results in a diminishing product:

• Ultimately, \(\left(\frac{4}{9}\right)^n\) will approach zero as \(n\) increases.

Thus, we conclude that the sequence converges to a limit of zero. Understanding limit evaluation provides insights into how sequences behave in the long run, which is a fundamental concept in calculus and mathematical analysis.