Prove that an absolutely convergent series \(\sum_{n-1}^{\infty} a_{n}\) is convergent. Hint: Put \(b_{n}=a_{n}+\left|a_{n}\right|\). Then the \(b_{n}\) are nonnegative; we have \(\left|b_{n}\right| \leq 2\left|a_{n}\right|\) and \(a_{n}=b_{n}-\left|a_{n}\right|\).

Understanding series convergence is crucial for many areas of mathematics. When we talk about a series \(\textstyle \sum_{n=1}^{\infty} a_{n}\) converging, we mean that the sequence of partial sums \(\textstyle S_N = \sum_{n=1}^{N} a_{n}\) approaches a finite limit as \(\textstyle N\) goes to infinity. In other words, as you add more terms, the total gets closer to a fixed number.
When we say a series \(\textstyle \sum_{n=1}^{\infty} a_{n}\) converges absolutely, it means that the series \(\textstyle \sum_{n=1}^{\infty} |a_{n}|\) also converges. Absolute convergence is a very strong form of convergence because it guarantees convergence even if we shuffle the order of terms.
In our exercise, we are given an absolutely convergent series and need to prove that the series itself is convergent. The step-by-step solution demonstrates that this is true. When you have absolute convergence, regular convergence follows as a consequence.

The comparison test is a handy tool in determining whether a series converges. It involves comparing the terms of your series to those of another series that you already know converges or diverges.
Here's how the comparison test works:

• Find a series \(\textstyle \sum b_n\) such that all terms \(\textstyle b_n\) are positive and known to converge.
• Ensure that your series' terms are smaller or equal to the terms of the known series \(\textstyle |a_n| \leq |b_n|\).

If both conditions are met, then your series \(\textstyle \sum a_n \) converges. In the given problem, we use the new sequence \(\textstyle b_n = a_n + |a_n|\) and show that \(\textstyle |b_n| \leq 2 |a_n|\). Knowing that \(\textstyle \sum 2|a_n| \) converges (from absolute convergence) allows us to conclude that \(\textstyle \sum b_n \) converges too.

A sequence is said to be bounded if there is a constant \(\textstyle M\) such that every term of the sequence is less than or equal to \(\textstyle M\). Bounded sequences are significant in the study of series because they help in proving convergence.
In our specific problem, the transformation \(\textstyle b_n = a_n + |a_n|\) produces a sequence \(\textstyle b_n\) that is nonnegative and bounded by \(\textstyle 2|a_n|\). This bounding is crucial because it enables the use of the comparison test.
Since \(\textstyle |b_n| \leq 2 |a_n|\), we know that the terms of \(\textstyle b_n\) are under control with respect to \(\textstyle 2|a_n|\). This bounded nature of \(\textstyle b_n\), combined with the known convergence of \(\textstyle \sum 2 |a_n|\), allows us to deduce the convergence of \(\textstyle \sum b_n\) and finally \(\textstyle \sum a_n\).