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Chapter 10: Problem 29

Limits of sequences Find the limit of the following sequences or determine that the sequence diverges. $$\left\\{(0.5)^{n}+3(0.75)^{n}\right\\}$$

Short Answer

Expert verified
Answer: The limit of the sequence is 0.

Step by step solution

01

Identify the type of sequence and its terms

We are given the sequence: $$a_n = (0.5)^n + 3(0.75)^n$$ It is an exponential sequence because both terms have a constant raised to the power of n.
02

Apply limit properties

We will apply the limit properties to the given sequence. Since it's a sum, we can take the limit of each term separately: $$\lim_{n \to \infty} a_n = \lim_{n \to \infty} (0.5)^n + \lim_{n \to \infty} 3(0.75)^n$$
03

Determine the behavior of each term as n approaches infinity

Now, analyze each term individually. 1. For the term \((0.5)^n\): Since the base is less than 1, as n approaches infinity, the term will approach 0. 2. For the term \(3(0.75)^n\): Since the base (0.75) is also less than 1, as n approaches infinity, the term will approach 0.
04

Calculate the limit of the sequence

Now, calculate the limit of the given sequence: $$\lim_{n \to \infty} a_n = \lim_{n \to \infty} (0.5)^n + \lim_{n \to \infty} 3(0.75)^n = 0 + 0 = 0$$ The limit of the given sequence is 0.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Sequence
An exponential sequence is a series where each term is formed by raising a constant base to the power of the sequence index, called \( n \). This type of sequence can often be represented as \( a^n \), where \( a \) is the base and has a significant effect on the sequence's behavior as \( n \) grows. Some key characteristics are:
  • The base \( a \) determines how the sequence grows or decays.
  • If \( a > 1 \), the sequence grows exponentially.
  • If \( 0 < a < 1 \), the sequence decays exponentially, heading towards zero.
In our exercise, the sequence is \((0.5)^n + 3(0.75)^n\). Each term has a base less than 1, indicating exponential decay. Understanding this helps predict how each part of the sequence behaves over time.
Sequence Convergence
Sequence convergence is about whether a sequence eventually settles to a particular number, known as the limit. To determine convergence:
  • Examine whether the sequence's terms get closer to a single value.
  • If it does, the sequence converges.
  • If not, the sequence is said to diverge.
In our example, we look at the expression \((0.5)^n + 3(0.75)^n\). As \( n \) approaches infinity, both terms \((0.5)^n\) and \(3(0.75)^n\) shrink towards zero.This means the sequence converges to the limit of 0. Recognizing convergence helps us understand the eventual behavior of sequences.
Limit Properties
Limit properties refer to a set of rules and tactics that help us calculate the limit of more complex sequences. These properties make evaluating limits more manageable:
  • The limit of a sum is the sum of the limits: \( \lim_{n \to \infty} (a_n + b_n) = \lim_{n \to \infty} a_n + \lim_{n \to \infty} b_n \).
  • The limit of a product is the product of the limits if each limit exists.
  • These properties allow breaking down sequences into simpler parts to find the limit.
For the sequence \((0.5)^n + 3(0.75)^n\), we evaluate each term separately. We apply the sum of limits property, calculating each term's limit as \( n \) approaches infinity and summing them for the final result.
Infinity Behavior
Infinity behavior involves understanding how a sequence acts as the index \( n \) tends toward infinity. Analyzing infinity behavior helps us predict long-term trends and the convergence of sequences.
  • If a term's base is less than 1, as \( n \to \infty \), it tends to zero.
  • If the base is exactly 1, the term stays constant.
  • Bases greater than 1 indicate unbounded growth.
For the given sequence, both \((0.5)^n\) and \(3(0.75)^n\) are terms with bases less than 1. So, as \( n \) increases, both terms diminish to 0. This reduces the complete sequence to the behavior seen at infinity, which in this problem is convergence to 0. Understanding infinity behavior is critical for predicting sequence conclusions.

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