Proving Divergence of Sequence to Negative Infinity

Prove that the sequence $a_n = -\frac{n}{2}$ diverges to negative infinity.

In this problem, you are tasked with proving that a given sequence, specifically an arithmetic sequence in this case, diverges to negative infinity. Understanding the divergence to negative infinity involves comprehending the behavior of sequences as their indices approach infinity. For the sequence provided, observe that each term is negative and decreases linearly without bound. Divergence occurs when the terms of the sequence decrease indefinitely, never settling into a finite limit.

To tackle this problem, consider the definition of divergence to negative infinity: for every real number, however negative, there exists a point in the sequence beyond which all terms are less than that number. Essentially, it involves demonstrating that the sequence continues to drop lower than any potential lower limit you consider, cementing the idea of an ever-descending set of terms within the sequence.

Applying these concepts will foster a deeper understanding of not only sequence behavior but also the meticulous nature of analyzing the boundless growth or decay of mathematical functions. Recognize that this type of sequence provides a clear example of divergence, aiding in the foundational understanding of how sequences interact with infinity and converge or diverge based on their specific patterns and formulations.