Problem 99 - Essential Calculus

Question

Suppose that \(\sum a_{n}\) and \(\sum b_{n}\) are series with positive terms. Prove that if \(\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=0\) and \(\sum b_{n}\) converges, \(\Sigma a_{n}\) also converges.

Step-by-Step Solution

Answer

Since \(a_n\) is ultimately less or equal a certain fraction of \(b_n\) and the series \(\sum b_n\) converges, then \(\sum a_n\) also converges. Therefore, given \(\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=0\) and \(\sum b_{n}\) converges, \(\sum a_{n}\) must also converge.

Step 1: Assert the Limit Condition

Given we know that \(\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=0\). This essentially means that for each \(\epsilon > 0\) there exists an \(N \in \mathbb{N}\) such that if \(n > N\) then \(|\frac{a_{n}}{b_{n}}| \leq \epsilon \). Instead of \(\epsilon\) we can use a positive number \(l\) satisfying \(0 < l < 1\). Hence, for all \(n > N\), \(\frac{a_{n}}{b_{n}} \leq l\) which implies that \(a_n \leq l * b_n\) for all \(n > N\).

Step 2: Apply Calculation

What \(a_n \leq l * b_n\) means that, for all \(n > N\), the \(n-th\) term of the \(a\) series is less than or equal to a fraction of corresponding \(n-th\) term of the \(b\) series. Given \(\sum b_n\) converges, it implies, by the comparison test that \(0 \leq a_n \leq l * b_n\) for all \(n > N\) also converges.

Step 3: Conclude The Proof

In conclusion, as per the conditions given in the problem, if \(\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=0\) and \(\sum b_{n}\) converges, then \(\sum a_{n}\) must converge. This is because we determined that \(a_n\) is ultimately less or equal a fraction \(l\) of \(b_n\), and since the series \(\sum b_n\) converges, then \(\sum a_n\) also converges by comparison test.

Key Concepts

Limit Comparison Test

The Limit Comparison Test is a powerful tool for determining the convergence of series, especially those with non-negative terms. The test involves comparing two series, say \(\sum a_n\) and \(\sum b_n\), to see if they behave similarly as \(n\) approaches infinity.
To apply this test, we first calculate the limit \(\lim_{n \to \infty} \frac{a_n}{b_n}\). If this limit evaluates to a positive finite number, the fate of \(\sum a_n\) and \(\sum b_n\) is tied together:

• If \(\sum b_n\) converges, then \(\sum a_n\) also converges.
• If \(\sum b_n\) diverges, then \(\sum a_n\) also diverges.

In the specific example we've examined where \(\lim_{n \to \infty} \frac{a_n}{b_n} = 0\), this implies \(a_n\) becomes significantly small compared to \(b_n\) as \(n\) becomes very large. Since we've established \(\sum b_n\) converges, using the Limit Comparison Test with this limit indicates that \(\sum a_n\) must also converge.

Convergent Series

When we refer to a convergent series, we're talking about a series \(\sum a_n\) where the sequence of partial sums approach a finite limit. In simpler terms, as you add more terms, the total sum approaches a specific number.
This is significant because convergent series behave predictably and have finite bounds. For a series of positive terms like \(\sum b_n\) to converge, its terms must not only get smaller but also get sufficiently smaller quickly enough as \(n\) increases.
Understanding convergence plays a crucial role when comparing series. If one series converges, we can often infer properties about a related series, as demonstrated in our example. Through comparison tests, such as the Limit Comparison Test, we can deduce the convergence of one series based on another already known to be convergent.

Series with Positive Terms

Series with positive terms are particularly manageable when determining convergence. This is because their partial sums either increase or remain constant, making their behavior easier to predict.
Unlike series that involve alternating signs, series with positive terms cannot cancel each other out, ensuring an upward or stable progression. For instance, in our main example, both \(\sum a_n\) and \(\sum b_n\) are series with positive terms. This allows us to apply comparison tests effectively without concern for sign alternations complicating the results.

• The Comparison Test and Limit Comparison Test are especially useful for series with positive terms.
• The non-negative nature ensures partial sums are simple to calculate and assess for convergence.

This simplicity is crucial in both theoretical and practical applications, enabling confident assessments about the convergence properties of a series.