Problem 79 - Calculus for Biology and Medicine

Question

Use the limit laws to determine \(\lim _{n \rightarrow \infty} a_{n}=a .\) $$ \lim _{n \rightarrow \infty}\left[\left(\frac{1}{3}\right)^{n}+\left(\frac{1}{2}\right)^{n}\right] $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Identify Each Part of the Limit Expression

The given limit expression is \( \lim_{n \rightarrow \infty} \left[ \left( \frac{1}{3} \right)^{n} + \left( \frac{1}{2} \right)^{n} \right] \). This is the sum of two separate sequences: \( \left( \frac{1}{3} \right)^{n} \) and \( \left( \frac{1}{2} \right)^{n} \).

Step 2: Analyze Individual Sequences

Consider each sequence separately:- For \( \left( \frac{1}{3} \right)^{n} \), as \( n \to \infty \), since the base \( \frac{1}{3} < 1 \), the sequence approaches 0.- For \( \left( \frac{1}{2} \right)^{n} \), as \( n \to \infty \), since the base \( \frac{1}{2} < 1 \), this sequence also approaches 0.

Step 3: Apply Limit Laws to the Sum

Use the limit law which states that the limit of a sum is the sum of the limits:\[ \lim_{n \rightarrow \infty} \left( \left( \frac{1}{3} \right)^{n} + \left( \frac{1}{2} \right)^{n} \right) = \lim_{n \rightarrow \infty} \left( \frac{1}{3} \right)^{n} + \lim_{n \rightarrow \infty} \left( \frac{1}{2} \right)^{n} \]Since both individual limits are 0, the combined limit is \( 0 + 0 = 0 \).

Step 4: State the Conclusion

Using the limit laws, we conclude that the limit of the given expression is 0:\[ \lim _{n \to \infty} \left( \left( \frac{1}{3} \right)^{n} + \left( \frac{1}{2} \right)^{n} \right) = 0 \]

Key Concepts

Sequences

A sequence is a list of numbers arranged in a specific order. Each number in the sequence is called a term. Understanding sequences involves recognizing patterns and determining the behavior of the sequence as it progresses.
In mathematical analysis, sequences often play a crucial role in exploring limits, as they allow us to analyze patterns and trends.

• Finite Sequences: A finite sequence has a specific number of terms. For example, the sequence 2, 4, 6, 8 is finite and consists of 4 terms.
• Infinite Sequences: An infinite sequence has no end; it continues indefinitely. The sequence 1, 1/2, 1/3, 1/4,... is infinite.

In the context of limits, we often deal with infinite sequences, analyzing what happens as the number of terms approaches infinity.
The behavior of each sequence can be very different but often follows recognizable patterns that can be managed through limit laws.

Infinite Limits

Infinite limits refer to the behavior of a function or sequence as it approaches infinity. In cases like the provided exercise, we are interested in how the terms behave as the index grows larger and larger.
When dealing with infinite sequences, such as \[a_n = \left(\frac{1}{3}\right)^{n} + \left(\frac{1}{2}\right)^{n}\], the purpose is to determine what value the sequence is approaching.
Key points about infinite limits:

• Convergence: If a sequence approaches a specific value as \(n\) tends to infinity, it is said to converge to that value.
• Divergence: If a sequence doesn’t approach any value and continues to grow indefinitely or oscillate, it diverges.

In this case, both parts of the sequence converge to 0, thanks to their exponents being negative and the bases less than 1. This leads to the sequence's limit also being 0, illustrating a common technique in mathematical analysis: separating and analyzing each component to understand the total behavior.

Mathematical Analysis

Mathematical analysis is a deep and fascinating field of mathematics that deals with limits, continuity, and the behavior of functions and sequences. It's critical for understanding complex mathematical concepts and real-world phenomena. When analyzing expressions like the one in the exercise, mathematical analysis provides tools to determine the behavior of sequences and functions.
One of the primary tools in mathematical analysis is the concept of a limit, which helps in understanding situations where direct computation isn't feasible. In our exercise, limit laws assisted in resolving the infinite sequence to 0 by examining each sequence component:

• Limit Laws: Useful rules that allow us to break down complex expressions. For example, the sum of limits is equal to the limit of the sums.
• Convergence and Divergence: Concepts that explain whether sequences settle at a certain value or not, respectively.

Overall, mathematical analysis plays a vital role in making sense of calculations involving infinity and helps pave the way for more advanced mathematical studies. Understanding these principles allows students to tackle a wide breadth of problems beyond simple arithmetic or algebraic manipulation.