Question

Suppose that \(\sum_{n=1}^{\infty} a_{n}=1\), that \(\sum_{n=1}^{\infty} b_{n}=-1\), that \(a_{1}=2\), and \(b_{1}=-3 .\) Find the sum of the indicated series. $$ \sum_{n=1}^{\infty}\left(a_{n}+b_{n}\right) $$

Short answer

The sum of the series is 0.

Step-by-step solution

Understanding the Problem

We need to find the sum of the series \( \sum_{n=1}^{\infty}(a_n + b_n) \). We know that \( \sum_{n=1}^{\infty} a_n = 1 \) and \( \sum_{n=1}^{\infty} b_n = -1 \). We also know that \( a_1 = 2 \) and \( b_1 = -3 \).

Key concepts

Convergent Series

In mathematics, an infinite series is termed as convergent if the sum of its terms approaches a specific value as the number of terms increases. This concept is foundational, as it helps in understanding whether the total of an infinite list of numbers can actually reach a finite sum.

To determine if a series is convergent, we primarily look at its terms and see if their sum stabilizes. For instance, consider our problem, where the series \( \sum_{n=1}^{\infty} a_n \) and \( \sum_{n=1}^{\infty} b_n \) each are convergent, with sums of 1 and -1, respectively. This means as more terms are added, the total closely approximates these values.

Understanding convergence is crucial, as it assures us about the behavior of an infinite series over time, something particularly important in every aspect of science and engineering.

Series Summation

Series summation involves adding up the terms of a sequence, which can be finite or infinite. This concept might initially seem straightforward, but infinite series requires a more profound approach since they involve an endless number of terms.

The given series \( \sum_{n=1}^{\infty}(a_n + b_n) \) can be deconstructed by leveraging the properties of series summation. Specifically, it relies on the idea that the sum of two series equals the sum of the individual sums. Therefore, knowing \( \sum_{n=1}^{\infty} a_n = 1 \) and \( \sum_{n=1}^{\infty} b_n = -1 \), we find:
- \( \sum_{n=1}^{\infty} (a_n + b_n) = \sum_{n=1}^{\infty} a_n + \sum_{n=1}^{\infty} b_n \)
- \( \sum_{n=1}^{\infty} (a_n + b_n) = 1 + (-1) = 0 \)

Thus, the series \( \sum_{n=1}^{\infty} (a_n + b_n) \) simply equals zero. It's a demonstration of how powerful and efficient series summation can be, reducing complex series to simple values by applying mathematical operations.

Mathematical Problem Solving

Problem-solving in mathematics is not only about calculating correct results but also developing a logical approach to understanding and simplifying problems. In this exercise, by recognizing known elements, we systematize the solving process efficiently.

To solve the problem, you first break it down: know your goals, what information is provided, and learn what relationships exist among elements like \( a_n \) and \( b_n \). Let's see how that unfolds here:

• Recognize the provided sums: \( \sum_{n=1}^{\infty} a_n = 1 \), \( \sum_{n=1}^{\infty} b_n = -1 \).
• Understanding the series structure helps: realize that \( a_1 = 2 \) and \( b_1 = -3 \) are merely individual terms and not directly needed for the sum of the whole series.
• Use simplification strategies by leveraging mathematical properties, like breaking down series into manageable components.

Through this methodical approach, solutions are not only found but are made understandable, emphasizing clarity in mathematical reasoning and the joy of simplifying complications.