Problem 14 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-20, an explicit formula for \(a_{n}\) is given. Write the first five terms of \(\left\\{a_{n}\right\\}\), determine whether the sequence converges or diverges, and, if it converges, find \(\lim _{n \rightarrow \infty} a_{n}\) \(a_{n}=\left(\frac{1}{4}\right)^{n}+3^{n / 2}\)

Step-by-Step Solution

Answer

The sequence diverges as \( 3^{n/2} \) grows without bound.

Step 1: Write first five terms

Compute \( a_1, a_2, a_3, a_4, a_5 \) from the given formula.

Step 2: Determine convergence

Since \( 3^{n/2} \) grows without bound, the sequence diverges.

Key Concepts

Explicit Formula

An explicit formula is a mathematical rule that provides a direct way to compute each term in a sequence based on its position, represented as "n." Unlike a recursive formula, where you need to know the previous term(s) to find the next one, the explicit formula lets you calculate any term directly.
This is particularly useful because it allows you to quickly find terms without referencing the entire sequence. For example, in the sequence described by \[a_{n} = \left(\frac{1}{4}\right)^n + 3^{n/2}\]you can determine any term by simply substituting the term number "n" into the formula.

The explicit formula saves time and effort, especially with large sequences, and is perfect for writing out the first few terms, analyzing the sequence's behavior, or determining if the sequence has certain properties like convergence.

Limit of a Sequence

The limit of a sequence describes the value that the terms of a sequence approach as the index "n" becomes very large, indicating the behavior of the sequence at infinity. When a sequence has a limit, we say that it "converges" to that limit.
In technical terms, for a sequence \(a_n\), the sequence converges to limit \(L\) if for every small number \( \varepsilon > 0\), there exists a number \(N\) such that for all \(n > N\), \( |a_n - L| < \varepsilon \). This condition ensures that the terms of the sequence get arbitrarily close to \(L\).

For the sequence given by the formula \[a_{n} = \left(\frac{1}{4}\right)^n + 3^{n/2},\]we can intuitively see that as \(n\) approaches infinity, the term \(3^{n/2}\) will grow much faster than \(\left(\frac{1}{4}\right)^n\) shrinks, suggesting that the sequence diverges.

Divergence

A sequence is said to diverge if it does not converge to a single finite limit. This means that as you take more terms of the sequence, they fail to approach a specific number. Divergence indicates that the terms of the sequence either increase infinitely, decrease without bound, or oscillate without settling to a single value.

For instance, examining the sequence \[a_{n} = \left(\frac{1}{4}\right)^n + 3^{n/2},\]one notices that the term \(3^{n/2}\) becomes very large as \(n\) increases, influencing the entire sequence to increase indefinitely. Here, the presence of \(3^{n/2}\) drives the divergence because its growth rate overpowers the diminishing effect of \(\left(\frac{1}{4}\right)^n\).

When solving problems involving divergence, it is crucial to determine which part of an expression dictates the sequence's behavior at infinity.

Sequence Terms

Terms in a sequence are the individual elements or numbers that make up the sequence. Each term has a specific position, often denoted by "n," which indicates its place in the sequence. Comprehending how each term is formed provides insight into the overall structure and pattern of the sequence.

For example, in the sequence defined by \[a_{n} = \left(\frac{1}{4}\right)^n + 3^{n/2},\]the terms are calculated by substituting values for \(n\). The first five terms are:- When \(n = 1\), the term \(a_1\) is \(\frac{1}{4} + \sqrt{3}\).- For \(n = 2\), the term \(a_2\) is \(\frac{1}{16} + 3\).- When \(n = 3\), the term \(a_3\) is \(\frac{1}{64} + \sqrt{27}\).- With \(n = 4\), the term \(a_4\) is \(\frac{1}{256} + 9\).- For \(n = 5\), the term \(a_5\) is \(\frac{1}{1024} + \sqrt{243}\).

Understanding these individual terms allows us to predict the sequence's pattern and analyze its behavior over time.