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Chapter 10: Problem 18

Determine whether the given sequence converges. $$ \left\\{4+\frac{3^{n}}{2^{n}}\right\\} $$

Short Answer

Expert verified
The sequence diverges, as the exponential term increases without bound.

Step by step solution

01

Review the Sequence

The given sequence is \( a_n = 4 + \frac{3^n}{2^n} \). We need to determine whether this sequence converges.
02

Simplify the Expression

Notice that the term \( \frac{3^n}{2^n} \) can be rewritten as \( \left(\frac{3}{2}\right)^n \). Therefore, the sequence becomes \( a_n = 4 + \left(\frac{3}{2}\right)^n \).
03

Analyze the Exponential Term

The term \( \left(\frac{3}{2}\right)^n \) is an exponential function with a base greater than 1. As \( n \rightarrow \infty \), this term will grow without bound, i.e., it tends to \( \infty \).
04

Determine the Convergence

Since \( \left(\frac{3}{2}\right)^n \rightarrow \infty \), the sequence \( a_n = 4 + \left(\frac{3}{2}\right)^n \) will also tend towards infinity. Therefore, the sequence diverges.

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Key Concepts

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

Exponential Sequences
Exponential sequences are a fascinating part of mathematics. They involve sequences where the terms are in the form of an exponential function.
In the exercise, the sequence involves an exponential term \( \left( \frac{3}{2} \right)^n \).
Such sequences can either grow very quickly or shrink based on the base value:
  • When the base is greater than 1, the terms increase rapidly, leading to divergence.
  • If the base is a fraction (less than 1), the terms decrease as \( n \) increases, possibly converging to 0.
Understanding exponential sequences is crucial as they can model various real-world phenomena like population growth and radioactive decay. Here, since our base \( \frac{3}{2} \) is greater than 1, the sequence grows exponentially.
Divergence of Sequences
Divergence is a critical concept in studying sequences. A sequence is said to diverge if it does not approach any specific limit as \( n \) increases indefinitely.
In our sequence \( a_n = 4 + \left( \frac{3}{2} \right)^n \), the sequence diverges because the term \( \left( \frac{3}{2} \right)^n \) grows without bound.
Divergent sequences do not settle down to a finite number, which means their limit is typically infinity. Some signs that a sequence is diverging include:
  • The terms increase without approximation to a boundary.
  • There is no consistent pattern in the values, preventing them from stabilizing.
Recognizing divergence helps in understanding the behavior and eventual destiny of sequences.
Limit of a Sequence
Understanding the limit of a sequence is foundational in calculus and mathematical analysis. The limit describes the value a sequence approaches as its term number () heads to infinity, if it exists.
Limits help determine whether a sequence converges or diverges. Converging sequences have a finite limit, whereas diverging ones do not.
In the exercise, the sequence \( a_n = 4 + \left( \frac{3}{2} \right)^n \) showed no sign of convergence:
  • The exponential term increased without bound, pushing the entire sequence towards infinity.
  • Therefore, no specific finite number could be the sequence's limit.
This exercise reinforces the awareness of limits, aiding in recognizing how sequences behave over time.

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Most popular questions from this chapter

The sequence $$ \left\\{1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}-\ln n\right\\} $$ is known to converge (albeit very slowly) to a number \(\gamma\) called Euler's constant. Calculate at least the first 10 terms of the sequence. Conjecture the limit of the sequence.

Suppose that \(p\) is the probability of an event occurring. Write a formula for the odds in favor of the event occurring. Write a formula for the odds against the event occurring.

Without adding the terms, determine the value of \(\sum_{k=0}^{4}\left(\begin{array}{l}4 \\ k\end{array}\right) 4^{k}\).

Use the Binomial Theorem to show that $$ \sum_{k=0}^{n}(-1)^{k}\left(\begin{array}{l} n \\ k \end{array}\right)=0 $$

Use one or more of the techniques discussed in this section to solve the given counting problem. A wine store has 12 different California wines and 8 different French wines. In how many ways can 6 bottles of wine consisting of 4 California and 2 French wines (a) be selected for display? (b) be placed in a row on a display shelf?
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