Problem 41 - Calculus: Early Transcendentals

Question

(a) Suppose that \( \sum a_n \) and \( \sum b_n \) are series with positive terms and \( \sum b_n \) is divergent. Prove that if \( \displaystyle \lim_{n \to \infty} \frac {a_n}{b_n} = \infty \) then \( \sum{a_n} \) is also divergent. (b) Use part (a) to show that the series diverges. (i) \( \displaystyle \sum_{n = 2}^{\infty} \frac {1}{\ln n } \) (ii) \( \displaystyle \sum_{n = 1}^{\infty} \frac {\ln n}{n} \)

Step-by-Step Solution

Answer

Both series diverge by the Limit Comparison Test.

Step 1: Understand the Limit Comparison Test

The Limit Comparison Test helps determine the convergence or divergence of two related series. For two series \( \sum a_n \) and \( \sum b_n \) with positive terms, if \( \lim_{n \to \infty} \frac{a_n}{b_n} = c \), where \( 0 < c < \infty \), both series either converge or diverge together.

Step 2: Prove Divergence of \( \sum a_n \) in Part (a)

Given that \( \lim_{n \to \infty} \frac {a_n}{b_n} = \infty \), this implies that \( a_n \) is eventually much larger than \( b_n \). If \( \sum b_n \) is divergent, by the Limit Comparison Test, \( \sum a_n \) must also be divergent because each term \( a_n \) becomes dominant compared to \( b_n \), leading to divergence.

Step 3: Examine Series in Part (b-i)

Consider \( \sum_{n=2}^{\infty} \frac{1}{\ln n} \). Choose \( b_n = \frac{1}{n} \), which is a known divergent harmonic series. Notice \( \lim_{n \to \infty} \frac{\frac{1}{\ln n}}{\frac{1}{n}} = \lim_{n \to \infty} \frac{n}{\ln n} = \infty \), confirming \( a_n \) grows much faster than \( b_n \). Thus, \( \sum_{n=2}^{\infty} \frac{1}{\ln n} \) diverges by the test.

Step 4: Examine Series in Part (b-ii)

Consider \( \sum_{n=1}^{\infty} \frac{\ln n}{n} \). Use \( b_n = \frac{1}{n} \), a divergent harmonic series. Calculate \( \lim_{n \to \infty} \frac{\frac{\ln n}{n}}{\frac{1}{n}} = \lim_{n \to \infty} \ln n = \infty \). As \( a_n \) grows significantly faster than \( b_n \), the series \( \sum_{n=1}^{\infty} \frac{\ln n}{n} \) diverges.

Key Concepts

Limit Comparison Test

The Limit Comparison Test is a powerful tool in calculus used to analyze the convergence or divergence of series. It is particularly useful when comparing two series with positive terms that behave similarly. The basic idea is to compare the terms of two series, say \( \sum a_n \) and \( \sum b_n \). If the limit \( \lim_{n \to \infty} \frac{a_n}{b_n} = c \) is a positive constant (\( 0 < c < \infty \)), then both series either converge or diverge together.Here's a scenario to better understand this test:

• If you are aware that \( \sum b_n \) converges or diverges, then knowing \( \lim_{n \to \infty} \frac{a_n}{b_n} = c \) allows you to conclude the same behavior for \( \sum a_n \).
• This comparison becomes incredibly beneficial when \( b_n \) is a well-known series like the harmonic series, which can provide an immediate reference for determination.

For a practical understanding, consider a situation where \( \lim_{n \to \infty} \frac{a_n}{b_n} = \infty \). This tells us that \( a_n \) grows faster than \( b_n \), thus if \( \sum b_n \) is divergent, \( \sum a_n \) must be divergent as well. By using this test, you can easily deduce divergence by comparing growth rates.

Convergence and Divergence

The terms convergence and divergence are used to describe the behavior of series as they extend towards infinity. A series converges if the sum of its terms approaches a finite number. Conversely, a series diverges if the sum grows without bound.Understanding these concepts is vital when performing series analysis:- **Convergence:** When looking at convergence, think of it as an endpoint that's eventually reached. For example, the geometric series \( \sum \left( \frac{1}{2} \right)^n \) converges to a finite limit.- **Divergence:** A series diverges if it doesn’t settle at a number. The harmonic series \( \sum \frac{1}{n} \) is a classic example. Despite individual terms diminishing, their sum trends towards infinity.In calculus, numerous tests help determine convergence or divergence of a series. This understanding is fundamental as it dictates the behavior of series in mathematical functions, greatly impacting how mathematicians and engineers solve problems.In essence, being able to establish whether a series converges or diverges provides critical insight into the behavior and sum of an infinite list of numbers.

Harmonic Series

The harmonic series is a well-known mathematical series that expresses the sum of reciprocals of all natural numbers: \( \sum \frac{1}{n} \). Its terms form a sequence like \( 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots \).What makes the harmonic series particularly famous is its divergence.

• At a glance, individual terms appear to get smaller rapidly, but the collective sum fails to converge.
• Despite the decrease in size of each reciprocal, the accumulation is slow yet persistent, resulting in divergence.

The divergence of the harmonic series contrasts with other series that converge regardless of diminishing terms. For instance, the latter halves of geometric series can converge to a sum due to exponentially shrinking terms.Why does this matter? Knowing this allows mathematicians to make educated guesses about convergence and divergence of other series by comparing them to the harmonic series. Its divergence often serves as a reference point or a basis of comparison within the Limit Comparison Test, thus acting as a bridge to understanding more complex series. This foundational knowledge enriches analysis, making it indispensable in higher mathematics.