Problem 57 - Thomas Calculus

Question

Suppose that \(a_{n}>0\) and \(b_{n}>0\) for \(n \geq N(N\) an integer). If \(\lim _{n \rightarrow \infty}\left(a_{n} / b_{n}\right)=\infty\) and \(\sum a_{n}\) converges, can anything be said about \(\sum b_{n}^{?}\) Give reasons for your answer.

Step-by-Step Solution

Answer

\( \sum b_n \) must diverge due to \( \lim_{n \rightarrow \infty} \frac{a_n}{b_n} = \infty \) given \( \sum a_n \) converges.

Step 1: Understand the given condition

We are given that \( \lim_{n \rightarrow \infty} \left( \frac{a_n}{b_n} \right) = \infty \). This means that as \( n \) becomes very large, \( a_n \) becomes much larger than \( b_n \). Also, \( \sum a_n \) is convergent.

Step 2: Analyze the implication of \( \lim_{n \rightarrow \infty} \left( \frac{a_n}{b_n} \right) = \infty \)

The condition \( \lim_{n \rightarrow \infty} \left( \frac{a_n}{b_n} \right) = \infty \) implies that for sufficiently large \( n \), the terms \( a_n \) dominate \( b_n \) significantly. Specifically, every large \( n \) will have \( a_n > kb_n \) for any fixed \( k > 0 \).

Step 3: Determine the behavior of \( \sum b_n \)

Since \( \sum a_n \) converges and \( a_n > kb_n \) for large \( n \), it follows that \( b_n \) must approach zero faster than a fixed multiple of \( a_n \). More intuitively, since \( a_n \) dominates \( b_n \) while keeping the series convergent, the series \( \sum b_n \) must be divergent to satisfy the given condition when \( a_n \) converges.

Step 4: Conclude with reasoning

Given \( \lim_{n \rightarrow \infty} \left( \frac{a_n}{b_n} \right) = \infty \) and \( \sum a_n \) converging, \( b_n \) must be zero in the limit (for convergence), implying the terms \( b_n \) must not vanish fast enough. Thus, \( \sum b_n \) diverges.

Key Concepts

Limit Comparison Test

The Limit Comparison Test is a powerful tool in determining the convergence or divergence of series. Imagine you have two series, say, \( \sum a_n \) and \( \sum b_n \). It's often useful to compare these two series to understand their behavior.
The test states that if \( a_n > 0 \) and \( b_n > 0 \) for all \( n \) sufficiently large, and if \( \lim_{n \to \infty} \frac{a_n}{b_n} = c \), where \( c \) is a positive constant, then both \( \sum a_n \) and \( \sum b_n \) either converge or diverge together.
- If \( c \) is a non-zero finite number, the behavior of \( \sum a_n \) gives us insight into \( \sum b_n \) and vice versa.- This test is particularly useful when direct comparison of terms seems difficult but the limit of their ratio is easier to evaluate.
In our specific problem, even though \( \lim_{n \rightarrow \infty} \left( \frac{a_n}{b_n} \right) = \infty \) does not give a finite constant, the Limit Comparison Test tells us about the growing behavior of the terms. Essentially, this means \( a_n \) grows much faster than \( b_n \) for large \( n \), which affects the divergence of \( \sum b_n \).

Divergent Series

A divergent series is one that does not sum up to a finite limit. Let's say we have a series \( \sum b_n \). If \( \sum b_n \) diverges, it implies adding up all the terms does not lead to a specific finite number. It might tend to infinity, oscillate, or behave erratically.
- Divergent series are warners that can suggest careless decisions in infinite sums.- Many natural mathematical concepts suggest divergence; for instance, the harmonic series is a common example of divergence.
In the context of our problem, we are told \( \sum a_n \) converges, yet \( \lim_{n \to \infty} \frac{a_n}{b_n} = \infty \). Here, \( a_n \) quite literarily dominates \( b_n \). Given this dominance and convergence, \( b_n \) itself cannot diminish rapidly enough to produce a convergent series. Therefore, it results in the series \( \sum b_n \) diverging.

Infinite Limits

Infinite limits usually indicate that as \( n \) grows larger, a sequence or function doesn't stabilize but keeps increasing or decreasing endlessly. In mathematical terms, \( \lim_{n \to \infty} f(n) = \infty \) means the values do not level off but continue in their trend toward infinity.
- Infinite limits play a crucial role in understanding series and their long-term behavior.- They often serve as indicators of the relative growth rates of sequences, especially when comparing them, like \( a_n \) and \( b_n \).
In our original exercise, since \( \lim_{n \to \infty} \left( \frac{a_n}{b_n} \right) = \infty \), it implies \( a_n \) increases much faster than \( b_n \) does as \( n \) becomes very large. This massive growth discrepancy makes it clear that even if \( \sum a_n \) converges, \( \sum b_n \), under such dominance, simply cannot keep pace and thus diverges. Understanding infinite limits helps predict the outcomes of such scenarios efficiently.