Problem 47 - Single Variable Calculus

Question

Suppose that \(\sum a_{n}\) and \(\Sigma b_{n}\) are convergent series with positive terms. Show that \(\sum a_{n} b_{n}\) is convergent. Hint: There exists an integer \(N\) such that \(n \geq N\) implies that \(b_{n} \leq 1\), and therefore, \(a_{n} b_{n} \leq a_{n}\) for \(n \geq N\)

Step-by-Step Solution

Answer

To show that the series \(\sum a_{n} b_{n}\) is convergent, we can follow these steps: 1. Find an integer \(N\) such that for \(n \geq N\), \(b_{n} \leq 1\), which is possible since \(\sum b_{n}\) is convergent with positive terms. 2. Show that for \(n \geq N\), \(a_{n} b_{n} \leq a_{n}\), since \(b_{n} \leq 1\). 3. Apply the comparison test for series. Since \(\sum a_{n}\) is convergent and \(a_{n} b_{n} \leq a_{n}\) for \(n \geq N\), \(\sum a_{n} b_{n}\) must also be convergent.

Step 1: Find the integer N that satisfies the hint condition

Since the series \(\sum b_{n}\) is convergent with positive terms, there must exist an integer \(N\) such that \(n \geq N\) implies \(b_{n} \leq 1\), as the terms of a convergent series must approach zero. We can find this integer \(N\) either by inspection or using the properties of convergence.

Step 2: Show that \(a_{n} b_{n} \leq a_{n}\) for \(n \geq N\)

Since we are given that for \(n \geq N\), \(b_{n} \leq 1\), we can multiply \(a_{n}\) by this inequality: \[a_{n}b_{n}\leq a_{n}\] This is true for all \(n \geq N\).

Step 3: Apply the comparison test

Now we can apply the comparison test for series. We have two series \(\sum a_{n}\) and \(\sum a_{n} b_{n}\) such that the partial sums satisfy \(a_{n} b_{n} \leq a_{n}\) for \(n \geq N\). Since \(\sum a_{n}\) is convergent (given in the problem statement), by the comparison test, \(\sum a_{n} b_{n}\) must also be convergent. Therefore, we have shown that if \(\sum a_{n}\) and \(\sum b_{n}\) are convergent series with positive terms, then the series \(\sum a_{n} b_{n}\) is also convergent.

Key Concepts

Convergent Series

In the realm of mathematics, particularly in the study of infinite sequences and series, it's essential to understand the concept of a convergent series. A series is simply the sum of the terms of a sequence. Now, when does it converge?

A convergent series is one where the sum of its terms approaches a finite number as more and more terms are added. In other words, as you progress along the series, adding term after term, the total sum doesn't shoot off to infinity but instead settles down, becoming closer and closer to a specific value. This behavior is indicative of a series that is well-behaved and predictable, which is invaluable in various fields of science and engineering.

Comparison Test

When dealing with series, particularly when trying to determine if they converge or diverge, the comparison test is an incredibly useful tool. It allows us to compare two series to each other to make this determination more straightforward.

Here’s how it works: if you have two series, one of which you know converges, and each term in your second series is equal to or smaller than the corresponding term in the known convergent series (and all terms are positive), then your second series will also converge. The comparison test relies on this concept of term-by-term comparison, and if the terms of the series you're testing are always less than the terms of a convergent series, you've got a convergent series on your hands too.

Series with Positive Terms

Delving into the specific type of series, we encounter series with positive terms. This means each term added to the sum is positive, never negative or zero. You may wonder, why does positivity matter?

Series with all positive terms have unique properties that influence their convergence. For instance, they cannot oscillate between values as series with negative terms might. Instead, they either grow without bound or approach a limit. This predictability is why such series are often easier to analyze for convergence, and why tests like the aforementioned comparison test apply so readily to them.

Properties of Convergence

Understanding the properties of convergence is vital for studying series. These properties help us determine whether an infinite series will settle down to a finite value or if it will continue to grow without bound. One fundamental property is that the terms of a convergent series must become smaller and smaller; they must approach zero as you take the sequence further and further out.

Another crucial property is that the convergence of a series is unaffected by adding, removing, or changing a finite number of terms. This emphasizes that it's the behavior of the series in the long run (as the number of terms goes to infinity) that really matters. These properties build a foundation that allows us to use various tests, like the comparison test, to analyze the convergence of series effectively and conclusively.