Problem 48 - Single Variable Calculus

Question

Suppose that \(\sum a_{n}\) is a convergent series with positive terms and \(\left\\{c_{n}\right\\}\) is a sequence of positive numbers that converges to zero. Prove that \(\Sigma a_{n} c_{n}\) is convergent. Hint: There exists an integer \(N\) such that \(n \geq N\) implies that \(c_{n}

Step-by-Step Solution

Answer

Given a convergent series with positive terms \(\sum a_n\) and a sequence of positive numbers \(\{c_n\}\) that converges to zero, we will apply the comparison test using the hint provided. Since there exists an integer \(N\) such that for any \(n\geq N\), we have \(c_n

Step 1: 1. Convergence of the series \(\sum a_n\)

Since \(\sum a_n\) is a converging series with positive terms, we can use the comparison test to study the convergence of other series. Let \(s_n\) be the partial sum, \(s_n=\sum_{k=1}^{n} a_k\).

Step 2: 2. Applying the given hint

According to the provided hint, there exists an integer \(N\) such that for any \(n\geq N\), it is true that \(c_n

Step 3: 3. Determine if \(\sum L a_n\) is convergent

The series \(\sum L a_n\) is simply a multiple of the convergent series \(\sum a_n\). Since \(\sum a_n\) is convergent and \(L\) is a constant positive number, \(\sum L a_n\) must be convergent as well.

Step 4: 4. Apply the comparison test

By the comparison test, since \(0 \leq a_n c_n < L a_n\) for \(n\geq N\) and \(\sum L a_n\) is convergent, it follows that the series \(\sum a_n c_n\) is also convergent.

Key Concepts

Comparison Test

Understanding the comparison test is pivotal when investigating the convergence of series with positive terms. The core idea is simple: if you have two series with positive terms, and the terms of one series are consistently smaller than the corresponding terms of another known convergent series, then the smaller series also converges.

To apply the comparison test properly, identify a convergent 'benchmark' series. Then, compare each term of the series you're studying to the corresponding term of the benchmark series. If the terms of the series in question are less than or equal to the benchmark series for all terms past a certain point, convergence is guaranteed.

Let's say you have a series \( \sum b_n \) you wish to test and a known convergent series \( \sum a_n \). If \( b_n \leq a_n \) for all \( n \geq N \), a certain natural number, and the series \( \sum a_n \) is convergent, then you can confidently declare that the series \( \sum b_n \) is convergent too.

Partial Sum

A partial sum is a means of coming to grips with a series incrementally. When you're looking at the sum of an infinite series, it can be intimidating to contemplate its total. But by breaking it down and considering the sum of the first 'n' terms, you create a sequence of partial sums.

For a series \( \sum a_n \), the nth partial sum \( s_n \) is the sum of the first 'n' terms: \( s_n = \sum_{k=1}^{n} a_k \). Tracking how these partial sums behave as 'n' increases is how you ascertain the series' convergence or divergence. If the sequence of partial sums tends toward a specific limit as 'n' goes to infinity, the series converges; if not, it diverges.

Thinking about partial sums is also useful when you apply the comparison test, as you look at the behavior of the partial sums from both the series in question and the known convergent series.

Series Convergence

Series convergence is a fundamental concept in calculus and analysis, especially when dealing with infinite series. An infinite series converges if the sequence of its partial sums approaches a finite limit. In other words, as you add up more and more terms, the total settles down to a certain value rather than growing indefinitely or oscillating without bound.

It's important not only to establish whether a series converges, but also to understand the rate and nature of its convergence. This has practical implications in various applications like numerical analysis, where the convergence of series affects computation accuracy and efficiency.

Different tests for convergence exist, tailored to series with specific characteristics. For positive term series, the comparison test is often used because it provides an intuitive and simple way to relate unknown series to those which are well-understood. However, for more complex series, other tests such as the ratio test or the integral test might be more appropriate.

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Problem 48

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Key Concepts

Comparison Test

Understanding the comparison test is pivotal when investigating the convergence of series with positive terms. The core idea is simple: if you have two series with positive terms, and the terms of one series are consistently smaller than the corresponding terms of another known convergent series, then the smaller series also converges.

To apply the comparison test properly, identify a convergent 'benchmark' series. Then, compare each term of the series you're studying to the corresponding term of the benchmark series. If the terms of the series in question are less than or equal to the benchmark series for all terms past a certain point, convergence is guaranteed.

Let's say you have a series \( \sum b_n \) you wish to test and a known convergent series \( \sum a_n \). If \( b_n \leq a_n \) for all \( n \geq N \), a certain natural number, and the series \( \sum a_n \) is convergent, then you can confidently declare that the series \( \sum b_n \) is convergent too.

Partial Sum

A partial sum is a means of coming to grips with a series incrementally. When you're looking at the sum of an infinite series, it can be intimidating to contemplate its total. But by breaking it down and considering the sum of the first 'n' terms, you create a sequence of partial sums.

For a series \( \sum a_n \), the nth partial sum \( s_n \) is the sum of the first 'n' terms: \( s_n = \sum_{k=1}^{n} a_k \). Tracking how these partial sums behave as 'n' increases is how you ascertain the series' convergence or divergence. If the sequence of partial sums tends toward a specific limit as 'n' goes to infinity, the series converges; if not, it diverges.

Thinking about partial sums is also useful when you apply the comparison test, as you look at the behavior of the partial sums from both the series in question and the known convergent series.

Series Convergence

Series convergence is a fundamental concept in calculus and analysis, especially when dealing with infinite series. An infinite series converges if the sequence of its partial sums approaches a finite limit. In other words, as you add up more and more terms, the total settles down to a certain value rather than growing indefinitely or oscillating without bound.

It's important not only to establish whether a series converges, but also to understand the rate and nature of its convergence. This has practical implications in various applications like numerical analysis, where the convergence of series affects computation accuracy and efficiency.

Different tests for convergence exist, tailored to series with specific characteristics. For positive term series, the comparison test is often used because it provides an intuitive and simple way to relate unknown series to those which are well-understood. However, for more complex series, other tests such as the ratio test or the integral test might be more appropriate.