Problem 65 - Thomas Calculus

Question

Assume that \(b_{n}\) is a sequence of positive numbers converging to 4\(/ 5 .\) Determine if the following series converge or diverge. $$ \begin{array}{ll}{\text { a. } \sum_{n=1}^{\infty}\left(b_{n}\right)^{1 / n}} & {\text { b. } \sum_{n=1}^{\infty}\left(\frac{5}{4}\right)^{n}\left(b_{n}\right)} \\ {\text { c. }} {\sum_{n=1}^{\infty}\left(b_{n}\right)^{n}} & {\text { d. } \sum_{n=1}^{\infty} \frac{1000^{n}}{n !+b_{n}}}\end{array} $$

Step-by-Step Solution

Answer

a. Diverges; b. Diverges; c. Converges; d. Converges.

Step 1: Evaluating series (a)

The series given is \(\sum_{n=1}^{\infty}\left(b_{n}\right)^{1 / n}\). Since \(b_{n} \to \frac{4}{5}\), it implies \(\left(b_{n}\right)^{1/n} \to 1\) as \(n \to \infty\). Thus, the terms \(\left(b_{n}\right)^{1/n}\) approach 1. By the divergence test, since the general term does not converge to 0, the series diverges.

Step 2: Evaluating series (b)

The series is \(\sum_{n=1}^{\infty}\left(\frac{5}{4}\right)^{n}\left(b_{n}\right)\). Considering \(b_{n}\to \frac{4}{5}\), for large \(n\), \(b_{n}\approx \frac{4}{5}\). Thus, \(\left(\frac{5}{4}\right)^{n}\left(b_{n}\right)\approx (1)^n = 1\), leads to the general term not approaching zero. By the divergence test, the series diverges.

Step 3: Evaluating series (c)

The series is \(\sum_{n=1}^{\infty}\left(b_{n}\right)^{n}\). As \(b_{n} \to \frac{4}{5} < 1\), we can apply the ratio test: \(\frac{a_{n+1}}{a_{n}} = \frac{\left(b_{n+1}\right)^{n+1}}{\left(b_n\right)^n} = b_{n+1} \to \frac{4}{5} < 1\). Therefore, the series converges according to the ratio test.

Step 4: Evaluating series (d)

The series is \(\sum_{n=1}^{\infty} \frac{1000^{n}}{n !+b_{n}}\). The behavior compared to \(\frac{1000^{n}}{n!}\) (which converges by the ratio test, as \(\frac{1000^{n+1}}{(n+1)!}\div\frac{1000^{n}}{n!} = \frac{1000}{n+1} \to 0\)). \(b_{n}\approx \frac{4}{5} > 0\) doesn't impact convergence and effectively reduces to the convergent series. Thus, this series converges.

Key Concepts

Sequence Limit

In mathematics, the concept of a sequence limit is central to understanding the behavior of sequences as they progress. A sequence \( \{b_n\} \) of numbers is said to converge to a limit \( L \) if, as \( n \) becomes very large, the terms of the sequence approach \( L \). In formal terms, for every positive number \( \epsilon \), there exists a natural number \( N \) such that if \( n > N \), then \( |b_n - L| < \epsilon \). In the exercise, the sequence is given with \( b_n \to \frac{4}{5} \), meaning as we look at larger and larger \( n \), the sequence values get closer to \( \frac{4}{5} \). Understanding limits helps us determine the behavior of functions and series derived from such sequences.

Divergence Test

The divergence test, also known as the nth-term test for divergence, is a simple yet powerful tool for determining whether a series doesn't converge. It states that if the limit of the general term \( a_n \) of a series \( \sum a_n \) is not zero, then the series must diverge. In mathematical notation, if \( \lim_{n \to \infty} a_n eq 0 \), the series \( \sum a_n \) diverges. This test was utilized in parts (a) and (b) of the exercise. In series (a), since \((b_n)^{1/n} \to 1\) and does not approach zero, the series diverges as per the divergence test. Likewise, in series (b), the terms do not approach zero, leading to divergence. It's crucial to remember though, while a term limit of zero doesn't guarantee convergence, a non-zero limit assures divergence.

Ratio Test

The ratio test is another important tool in determining the convergence of an infinite series. It involves taking the limit of the absolute ratio of consecutive terms, \( \frac{a_{n+1}}{a_n} \). If this limit is less than one, the series will absolutely converge. If it is greater than one or infinity, the series diverges. And if the limit equals one, the test is inconclusive. In the given exercise for part (c), using the ratio test reveals the series \( \sum_{n=1}^\infty (b_n)^n \) converges. This is because \( b_n \to \frac{4}{5} \), resulting in the ratio \( \frac{a_{n+1}}{a_n} = b_{n+1} \to \frac{4}{5} \) which is less than one. Thus, the series converges, providing a clear case of how the ratio test aids in understanding series behavior.

Convergent Series

When a series has a sum that approaches a finite limit as more terms are added, it is called a convergent series. Convergent series are key in mathematics, allowing us to approximate values and solve complex problems. In mathematical terms, a series \( \sum a_n \) is convergent if the sequence of its partial sums \( S_n = a_1 + a_2 + ... + a_n \) approaches a finite number as \( n \to \infty \). Part (d) of the exercise shows how the series \( \sum \frac{1000^n}{n! + b_n} \) converges by comparing it to the convergent series \( \sum \frac{1000^n}{n!} \). Since adding \( b_n \), which converges to \( \frac{4}{5} \) and is negligible at large \( n \), does not disrupt the convergence, the whole series also converges. This illustrates an application of understanding how convergent series behave, an essential aspect of advanced mathematical studies.