Question

Suppose \(\sum_{n=0}^{\infty} a_{n}\) is convergent, and there are sequences \(\left\\{b_{n}\right\\}\) and \(\left\\{c_{n}\right\\}\) such that \(0 \leq b_{n} \leq a_{n} \leq c_{n}\) for all \(n\). What can be said about the series \(\sum_{n=0}^{\infty} b_{n}\) and \(\sum_{n=0}^{\infty} c_{n} ?\)

Short answer

\(\sum_{n=0}^{\infty} b_n\) converges; \(\sum_{n=0}^{\infty} c_n\)'s convergence is uncertain.

Step-by-step solution

Understand Given Information

We are given a series \(\sum_{n=0}^{\infty} a_{n}\) which is convergent. Also, there are sequences \(\{b_n\}\) and \(\{c_n\}\) with inequality relations: \(0 \leq b_{n} \leq a_{n} \leq c_{n}\) for all \(n\).

Analyze Sequence \(b_n\)

Since \(0 \leq b_{n} \leq a_{n}\) and \(\sum_{n=0}^{\infty} a_n\) converges, by the Comparison Test, \(\sum_{n=0}^{\infty} b_n\) also converges, as \(b_n\) is bounded above by a convergent series.

Analyze Sequence \(c_n\)

Since \(a_{n} \leq c_{n}\), we cannot immediately conclude about the convergence of \(\sum_{n=0}^{\infty} c_n\). If we find a bounded or convergent upper bound for \(c_n\), we can use the Comparison Test to infer convergence but the given information alone is insufficient for conclusively determining its convergence.

State Result Using Comparison Test

\(\sum_{n=0}^{\infty} b_n\) must converge because it is bounded by \(\sum_{n=0}^{\infty} a_n\), a convergent series. The convergence of \(\sum_{n=0}^{\infty} c_n\) cannot be determined conclusively as convergent since it could be diverging given it bounds \(a_n\) from above without additional conditions.

Key concepts

Convergent Series

A convergent series is one where the sum of its terms approaches a specific number, known as the limit, as more and more terms are added. In mathematical terms, if \( \sum_{n=0}^{\infty} a_{n} \) is a series, it converges if there is a finite limit \( L \) such that the partial sums \( S_N = a_0 + a_1 + a_2 + \ldots + a_N \) satisfy \( \lim_{N \to \infty} S_N = L \).

Convergence implies that as \( N \) becomes large, adding more terms results in smaller and smaller changes to the total sum. Understanding convergent series helps us determine the behavior of more complex sums and sequences, particularly in calculus and mathematical analysis.

• A series can only be convergent if its sequence of terms tends to zero as \( n \to \infty \). This is a necessary condition, but not sufficient on its own for convergence.
• Famous convergent series include the geometric series \( \sum_{n=0}^{\infty} r^n \), which converges for \( |r| < 1 \).

Sequence Inequality

The sequence inequality is the backbone of many comparison-based convergence tests. It refers to the order or relation between different sequences. In our context, we work with sequences \( \{b_n\} \), \( \{a_n\} \), and \( \{c_n\} \) that satisfy the condition \( 0 \leq b_{n} \leq a_{n} \leq c_{n} \) for all \( n \). This inequality helps us compare the behavior of the series formed by these sequences.

In simpler terms:

• The sequence \( \{b_n\} \) is always less than or equal to \( \{a_n\} \).
• The sequence \( \{a_n\} \) is less than or equal to \( \{c_n\} \).
• Both \( \{b_n\} \) and \( \{c_n\} \) are non-negative because they start from zero.

This inequality is crucial because it sets the stage for using the Comparison Test to determine convergence properties of the series built from these sequences.

Series Convergence Test

The Series Convergence Test, particularly the Comparison Test, is a tool used to determine if a series converges. When you have a known convergent series, you can use it to infer about the convergence of other series by comparing their terms.

For the Comparison Test, the principle is simple:

• If \( 0 \leq b_n \leq a_n \) for all \( n \), and \( \sum_{n=0}^{\infty} a_n \) converges, then \( \sum_{n=0}^{\infty} b_n \) also converges.
• Conversely, if \( a_n \leq c_n \), the convergence of \( \sum_{n=0}^{\infty} c_n \) cannot be assured without further information, even if \( \sum_{n=0}^{\infty} a_n \) is known to converge.

This test works because if a series is convergent, it means its terms become very small, summing to a finite value. Therefore, any smaller terms (like those in \( b_n \)) will also sum to a finite value. However, terms larger than \( a_n \), such as \( c_n \), might still diverge without additional constraints or information, despite the convergence of \( a_n \).