Problem 50 - Thomas Calculus

Question

Recursively Defined Terms Which of the series \(\sum_{n=1}^{\infty} a_{n}\) defined by the formulas in Exercises \(47-56\) converge, and which diverge? Give reasons for your answers. $$ a_{1}=3, \quad a_{n+1}=\frac{n}{n+1} a_{n} $$

Step-by-Step Solution

Answer

The series diverges because it is a multiple of the harmonic series.

Step 1: Determine the General Formula for Terms

We start with the recursive relation given: \(a_{1} = 3\) and \(a_{n+1} = \frac{n}{n+1} a_{n}\). Calculate the next few terms to understand the pattern:- \(a_{2} = \frac{1}{2} a_{1} = \frac{3}{2}\)- \(a_{3} = \frac{2}{3} a_{2} = \frac{2}{3} \cdot \frac{3}{2} = 1\)- \(a_{4} = \frac{3}{4} a_{3} = \frac{3}{4}\)From this pattern, we can express \(a_{n}\) as \(a_{n} = \frac{3}{n}\) for \(n \geq 1\).

Step 2: Evaluate the Convergence of the Series

Now consider the series \(\sum_{n=1}^{\infty} a_{n} = \sum_{n=1}^{\infty} \frac{3}{n}\). This series is a multiple of the harmonic series \(\sum_{n=1}^{\infty} \frac{1}{n}\), which is known to diverge. Therefore, the series \(\sum_{n=1}^{\infty} \frac{3}{n}\) also diverges.

Key Concepts

Convergence of a Series

Convergence in mathematics refers to a series approaching a specific limit or value as the number of terms increases. For a series \(\sum_{n=1}^{\infty} a_n\), to be convergent, its partial sums \(S_n = a_1 + a_2 + ... + a_n\) must approach a finite limit \(L\) as \(n\) goes to infinity. Several tests can help determine convergence, such as the comparison test, ratio test, or the root test. Typically, a series is considered convergent if the terms diminish sufficiently fast. This means that as \(n\) becomes very large, \(a_n\) should approach zero more rapidly. When working with recursive sequences like \(a_{n+1} = \frac{n}{n+1} a_n\), it's critical to analyze the pattern of the terms first. This helps assess whether they shrink at a rate that suggests convergence.

Divergence of a Series

Divergence indicates that a series does not have a finite limit as the number of terms increases. Instead of settling toward a specific number, the series \(\sum_{n=1}^{\infty} a_n\) either increases toward infinity or oscillates without approaching any particular value.A useful indicator of divergence for an infinite series is when its terms \(a_n\) do not approach zero. This is because if the terms remain significant in size, their contributions to the series continue indefinitely, preventing convergence to a finite value.In our example, the series \(\sum_{n=1}^{\infty} \frac{3}{n}\) is a modification of the harmonic series. Known for its divergence, the harmonic series \(\sum_{n=1}^{\infty} \frac{1}{n}\) fails to settle to a fixed number despite the individual terms reducing toward zero. Hence, any constant multiple of it, like \(\sum_{n=1}^{\infty} \frac{3}{n}\), also diverges.

Harmonic Series and Its Properties

The harmonic series is one of the most famous examples of a divergent series. It is defined as \(\sum_{n=1}^{\infty} \frac{1}{n}\). Despite each term becoming smaller as \(n\) increases, the series does not converge to a finite number.One way to understand the divergence of the harmonic series is through the integral test. By comparing it to the integral \(\int_{1}^{\infty} \frac{1}{x} \, dx\), which diverges, we conclude that the harmonic series also diverges.Additionally, although the terms \(\frac{1}{n}\) become small as \(n\) becomes large, they do not decline rapidly enough to approach a limit. This unique characteristic makes the harmonic series particularly interesting, demonstrating how size reduction alone is insufficient for convergence. The harmonic series serves as a perfect example of the divergence concept, emphasizing the importance of how terms shrink.