91Ó°ÊÓ

The Comparison Test is a powerful mathematical tool used to determine the convergence or divergence of an infinite series. It involves comparing a given series with another series whose convergence behavior is already known. The basic idea is straightforward:

• If the terms of the given series are smaller than the terms of a known convergent series and all terms are non-negative, the original series converges.
• Conversely, if the terms of the original series are larger than those of a known divergent series, it diverges.

This test is particularly useful because it allows us to leverage previously solved series to make conclusions about new, sometimes more complex expressions.
In the exercise, we use the Comparison Test by proving that each term of our series \( \frac{1}{n!} \) is less than or equal to the terms of the comparison series \( \frac{1}{n(n-1)} \). The comparison series is known to be convergent, thereby proving the convergence of the original series.

A telescoping series is a series where most terms cancel during summation. When the terms cancel, only a few terms from the beginning and the end of the series remain, greatly simplifying the sum.
An example of a telescoping series is \( \sum_{n=2}^{\infty} \left( \frac{1}{n-1} - \frac{1}{n} \right) \). When expanded, this series's terms cancel in succession:

• \( \left( \frac{1}{1} - \frac{1}{2} \right) + \left( \frac{1}{2} - \frac{1}{3} \right) + \left( \frac{1}{3} - \frac{1}{4} \right) + \cdots \)
• Almost every term in the series after the \( \frac{1}{1} \) term cancels out with its corresponding pair, leaving us with only the first fraction and the last in the expanded sequence.

Since the remaining terms do not add up infinitely, the series converges. Understanding the telescope effect in series helps in deducing them quickly and efficiently.

The factorial of a non-negative integer \( n \), denoted as \( n! \), is the product of all positive integers less than or equal to \( n \). Factorials grow very fast as \( n \) increases, illustrated as follows:

• \( 0! = 1 \)
• \( 1! = 1 \)
• \( 2! = 2 \times 1 = 2 \)
• \( 3! = 3 \times 2 \times 1 = 6 \)
• \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)

This rapid growth of factorial terms leads to the diminishing of terms like \( \frac{1}{n!} \), as \( n \) increases, tending to zero very quickly. This property makes factorial-based series candidates for convergence because large values of \( n \) create minor terms in the series.
Factorials are fundamental in combinatorial mathematics, with applications ranging from evaluating permutations to solving sequence problems, making them an essential part of mathematical studies and analyses.