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A sequence is a set of numbers arranged in a specific order. The numbers in a sequence are called its terms. Sequences can be finite or infinite, depending on whether they have a limited or limitless number of terms, respectively. A series is the sum of the terms of a sequence.

For instance, in the given exercise, the sequence defined by the formula \( a_n = \dfrac{1}{(n + 1)!} \) is infinite as we can keep calculating terms \( a_n \) for ever-increasing values of \( n \) without an end. The first five terms serve to illustrate the beginning of this sequence, showcasing how the formula is applied at different values of \( n \) to generate the sequence. It’s an exciting exploration into how mathematical expressions can create a patterned list of numbers that can potentially continue indefinitely.

The exclamation mark (!) in mathematics signifies 'factorial', a concept foundational to combinatorics, probability, and algebra. The factorial of a non-negative integer \( n \) is the product of all positive integers less than or equal to \( n \) and can be formally defined as:
\[ n! = n \times (n-1) \times (n-2) \times \cdots \times 3 \times 2 \times 1 \]
By convention, the factorial of zero (0!) is 1. Factorial notation is essential in the study of permutations and combinations, and in sequences like the one in our exercise where terms are defined using factorials. It’s a concept that intertwines multiplication with a decreasing sequence of natural numbers and has fascinating properties and applications across different areas of mathematics.

An arithmetic sequence, or arithmetic progression, is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the 'common difference'. An example of an arithmetic sequence is \( 2, 4, 6, 8, \ldots \), where each term after the first is obtained by adding 2.

Our current sequence, however, is not arithmetic since the difference between consecutive terms does not remain constant but instead depends on the factorial of the succeeding number. Understanding the distinction between different types of sequences helps students recognize patterns and apply the appropriate formulas to solve problems effectively.

The concept of convergence is crucial when it comes to sequences. A sequence converges if its terms approach a specific value as \( n \) increases without bound. This specific value is called the limit of the sequence. If a sequence does not converge, it is said to diverge.

In the context of our exercise, we can investigate the convergence by examining the behavior of the sequence as \( n \) gets very large. Since factorials grow at an extremely fast rate, the terms of the sequence \( a_n = \dfrac{1}{(n + 1)!} \) will get smaller and smaller, approaching zero. This means our sequence is a convergent sequence, with its limit being zero. Knowing about convergence and limits can help students not only to understand sequences better but also to dive into more advanced mathematical topics like calculus.