Mathematical proofs

Mathematical proofs are logical arguments that demonstrate the truth or validity of a mathematical statement or theorem. A proof provides an explanation of why a particular statement is true based on axioms (fundamental truths accepted without proof), previously established theorems, and logical reasoning. Key features of mathematical proofs include: 1. **Logical Structure**: A proof is constructed using a clear logical framework, often consisting of statements and arguments that follow a structured approach.

"Article proofs" typically refer to a stage in the academic publishing process where authors are provided with a formatted version of their manuscript, which is often referred to as a proof or galley proof. This version includes all the editorial revisions made after the original manuscript submission and allows authors to review the final layout, check for any typographical errors, and ensure that their work is accurately represented before the article is published in a journal.

"Articles containing proofs" typically refers to scholarly or academic articles that present formal proof for theorems or propositions in various fields, such as mathematics, computer science, logic, and statistics. These articles usually include a detailed explanation of the problem being addressed, the methodology used, and step-by-step reasoning leading to the conclusion.

A geometric series is a series of terms that have a constant ratio between successive terms. It is formed from a geometric sequence, which is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

The series given is a geometric series where the first term \( a \) is \( \frac{1}{2} \) and the common ratio \( r \) is \( \frac{1}{2} \).

We can represent the series \( S = \frac{1}{2} - \frac{1}{4} + \frac{1}{8} - \frac{1}{16} + \cdots \) more clearly by recognizing it as an infinite geometric series. ### Step 1: Identify the First Term and the Common Ratio The first term \( a \) of the series is \( \frac{1}{2} \).