Problem 88 - Thomas Calculus

Question

If \(\sum a_{n}\) converges and \(\sum b_{n}\) diverges, can anything be said about their term-by-term sum \(\sum\left(a_{n}+b_{n}\right) ?\) Give reasons for your answer.

Step-by-Step Solution

Answer

\(\sum (a_n + b_n)\) diverges because \(\sum b_n\) diverges.

Step 1: Understand the Series

We are given two series, \(\sum a_n\) which converges, and \(\sum b_n\) which diverges. The question asks about the convergence of \(\sum (a_n + b_n)\). To solve this, we need to evaluate how summing these series affects convergence.

Step 2: Review Convergence and Divergence

A series \(\sum a_n\) converges if the sequence of partial sums \(S_n = a_1 + a_2 + \ldots + a_n\) approaches a finite limit as \(n\rightarrow\infty\). On the other hand, \(\sum b_n\) diverges, meaning its sequence of partial sums does not approach a finite limit.

Step 3: Consider the Sum of Series

When two series \(\sum a_n\) and \(\sum b_n\) are added together term-by-term, we have \(\sum (a_n + b_n)\). The partial sums for this combination are given by \(S_n = (a_1 + b_1) + (a_2 + b_2) + \ldots + (a_n + b_n)\).

Step 4: Add Partial Sums

The partial sum \(S_n\) can be separated as \(S_n = (a_1 + a_2 + \ldots + a_n) + (b_1 + b_2 + \ldots + b_n)\). Due to the properties of series, this is equivalent to \(S_n = A_n + B_n\), where \(A_n\) is the partial sum of \(\sum a_n\) and \(B_n\) is the partial sum of \(\sum b_n\).

Step 5: Evaluate the Convergence of the Combined Series

Since \(A_n\) converges to a finite limit and \(B_n\) diverges, \(A_n + B_n\) will generally reflect the behavior of the divergent series \(B_n\), causing \(S_n\) to diverge. The divergent behavior typically dominates the summation.

Step 6: Conclusion

The series \(\sum (a_n + b_n)\) will diverge because \(\sum b_n\) diverges and adds unbounded values to \(\sum a_n\), making the combined series diverge.

Key Concepts

Convergent Series

A convergent series is a series where the sum of its terms approaches a specific, finite number as more terms are added. This concept is vital for understanding sequences and series in mathematics.
A series \( \sum a_n \) converges if the sequence of its partial sums, \( S_n = a_1 + a_2 + \ldots + a_n \), approaches a finite limit as \( n \to \infty \).
For example, the series \( \sum \frac{1}{n^2} \) converges because as you add more terms in this series, the sum starts stabilizing at a finite number, contributing less and less with each additional term. Convergent series are important because they allow us to assign a value to the sum of infinitely many terms, providing a sense of completeness in calculations.
Some key points about convergent series:

• The behavior of its terms: If \( a_n \to 0 \) as \( n \to \infty \), it suggests but does not guarantee convergence.
• Convergence tests: Tests like the Ratio Test or Integral Test help confirm if a series converges.

Divergent Series

A divergent series is one where the sum of its terms does not approach a specific, finite number, even as more terms are added. Unlike convergent series, divergent series tend to either increase indefinitely or oscillate without approaching any particular limit.
For a series \( \sum b_n \) to diverge, the sequence of partial sums \( B_n = b_1 + b_2 + \ldots + b_n \) continues to change as \( n \to \infty \), without settling down to a single value.
One familiar example of a divergent series is \( \sum \frac{1}{n} \), known as the harmonic series. Regardless of the number of terms summed, no finite value is reached.
Key characteristics of divergent series include:

• Non-zero term limit: If \( b_n ot\to 0 \) as \( n \to \infty \), the series certainly diverges.
• Absence of convergence: When no convergence test is satisfied, the series diverges.

Term-by-Term Addition

Term-by-term addition is a method of forming a new series by adding corresponding terms from two existing series. This process affects the convergence behavior of the resulting series.Suppose you have two series: \( \sum a_n \) that converges, and \( \sum b_n \) which diverges. When you add these series term-by-term, you form a new series, \( \sum(a_n + b_n) \), where each term is the sum of the corresponding terms from \( \sum a_n \) and \( \sum b_n \).
The partial sums of this new series can be expressed as \( S_n = (a_1 + b_1) + (a_2 + b_2) + \ldots + (a_n + b_n) \), which simplifies to the combined sums of both series' partial sums: \( S_n = A_n + B_n \).
In most cases, if one of the series is divergent, like \( \sum b_n \), the divergent behavior will dominate, and the new series \( \sum(a_n + b_n) \) will also diverge. Adding a divergent value even to a convergent sum almost always results in a divergent series. Understanding the nature of the individual series is crucial when performing term-by-term addition.