Created on July 10, 2023

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Summary: This blog post highlights the critical role of absolute convergence in mathematical analysis, exploring theorems and providing examples to illustrate its significance in ensuring reliable and consistent operations on infinite series or integrals.

Absolute convergence is critical in the field of mathematical analysis because it ensures that certain operations and manipulations involving infinite series or integrals can be performed reliably and consistently. It provides a rigorous foundation for analyzing and manipulating mathematical expressions and plays a fundamental role in many areas of mathematics, including calculus, real analysis, and complex analysis.

Let's look into some examples/theorems of absolute convergence.

$\textbf{ Theorem} \sum a_n \text { converges absolutely, then } \sum a_n \text { converges. }$

We know that for every $\epsilon >0$, we will have $M$ such that $\sum_{i=n}^m |a_i| < \epsilon$ for $n,m \ge M$. We have the following:

$$\left| \sum_{i=n}^m a_i\right| \le \sum_{i=n}^m |a_i| < \epsilon$$

So $\sum a_n$ converges.

For non converging absolute series, we can see the following example

$$\sum \frac{(-1)^n}{2n+1}=\frac{1}{4} (\pi-4)$$

though the series converges to a real number but the series does not converges absolutely