Problem 50 - Thomas Calculus

Question

Recursively Defined Terms Which of the series \(\sum_{n=1}^{\infty} a_{n}\) defined by the formulas in Exercises \(45-54\) converge, and which diverge? Give reasons for your answers. $$a_{1}=5, \quad a_{n+1}=\frac{\sqrt[n]{n}}{2} a_{n}$$

Step-by-Step Solution

Answer

The series converges, as its terms approach zero exponentially.

Step 1: Identify the Type of Series

The series in the problem is defined recursively, which means each term in the sequence depends on the previous terms. Specifically, the first term is defined as \(a_1 = 5\), and subsequent terms are defined by the formula \(a_{n+1} = \frac{\sqrt[n]{n}}{2} a_{n}\).

Step 2: Observe the Behavior of Recursive Formula

To analyze convergence or divergence, observe the behavior of the recursive formula \(a_{n+1} = \frac{\sqrt[n]{n}}{2} a_{n}\). Since \(\sqrt[n]{n}\) approaches 1 as \(n\) approaches infinity, the factor in the recursive formula becomes gradually \(\frac{1}{2}\). Therefore, each term in the sequence might be approximated as \(a_{n+1} \approx \frac{1}{2} a_{n}\) for large \(n\).

Step 3: Determine General Behavior of Terms

Given that each term approximately halves the previous term, the sequence will display a geometric-like behavior where each successive term is about half the size of the preceding one. Thus, the sequence will approach zero geometrically as the number of terms increases.

Step 4: Apply the Convergence Criteria

For the series to converge, we need the terms \(a_n\) to approach zero. Based on our analysis, the terms of the series approach zero because they decrease geometrically, like \(\frac{1}{2^n}\). The geometric series with ratio \(r = \frac{1}{2}\) indeed converges.

Key Concepts

Recursive Sequences

In mathematics, a recursive sequence is a sequence of numbers where each term is defined based on its previous terms. Such sequences can be helpful to model situations where current values depend on past ones. In our problem, we have a recursive definition:

• Start with a first term: \(a_1 = 5\)
• Define the next terms as \(a_{n+1} = \frac{\sqrt[n]{n}}{2} a_{n}\)

Here, each term \(a_{n+1}\) is obtained by multiplying the previous term \(a_n\) by the expression \(\frac{\sqrt[n]{n}}{2}\). As sequences progress, one key aspect to explore is how the terms evolve and whether they have a specific pattern.
Recursive sequences often involve mathematical investigation to understand long-term behavior. This can include checking if the sequence stabilizes to a fixed number, starts oscillating, or grows without limit. All these behaviors affect the convergence of the series formed by the sequence terms.

Geometric Series

A geometric series is a series of the form \(\sum_{n=0}^{\infty} ar^n \), where \(a\) is the first term and \(r\) is the common ratio. In our analysis, the sequence resembles a geometric pattern, especially as terms progress. Each term reduces in a manner similar to multiplying by a constant ratio. As \(n\) increases, \(\sqrt[n]{n}\) tends to 1, making our formula for \(a_{n+1}\) approach \(\frac{1}{2}a_{n}\). Thus, the sequence behaves like a geometric progression with a common ratio of approximately \(\frac{1}{2}\).

• When the ratio \(|r| < 1\), the geometric series converges.
• If \(|r| \geq 1\), the series diverges.

This concept is crucial for understanding how the formula affects the series' behavior over time and determining if the series will add up to a finite number or keep growing indefinitely.

Convergence Criteria

The convergence of a series refers to whether the sum of all its terms approaches a finite number. There are several criteria for testing convergence. For a series consisting of monotonically decreasing positive terms, commonly used convergence tests include:

• Geometric series test: If the ratio \(r\) of a series satisfies \(|r| < 1\), the series converges.
• Integral test, Ratio test, and Root test: These are standard tests based on comparing the series to an integral, or examining the limit of ratios and roots, respectively.

In our problem, since the sequence behaves as a geometric series with a ratio of approximately \(\frac{1}{2}\), the geometric series test confirms convergence.
Thus, as each term becomes progressively smaller, adding these diminishing numbers sums to a finite total. In simple words, the convergence criteria help us predict if endlessly adding the sequence's terms will result in a specific number.