91Ó°ÊÓ

When studying sequences, understanding their limit is crucial. A sequence \( \{a_n\} \) consists of a list of numbers, each corresponding to a natural number \( n \). Intuitively, the limit of a sequence refers to the value the terms of the sequence approach as \( n \) becomes very large. If a sequence approaches a specific value, it is said to converge, and that specific value is the limit.
In the given sequence \( \{a_n\} \) where \( a_n = \frac{(-4)^n}{n!} \), we analyze the behavior as \( n \) approaches infinity. This involves evaluating the expression for extremely large \( n \).
The concept of a limit helps determine if a sequence converges or diverges:

• If it converges, it means the terms settle around a particular number, which is its limit.
• Conversely, if it diverges, the sequence doesn't settle around any number and may approach infinity or oscillate indefinitely.

For \( a_n = \frac{(-4)^n}{n!} \), the sequence converges to zero because the denominator grows much faster than the numerator.

Factorial growth is a type of rapid growth that occurs when numbers are multiplied progressively larger integers. The factorial of a number \( n \), denoted as \( n! \), is the product of all positive integers less than or equal to \( n \). This characteristic leads to extremely fast growth rates.
For instance, consider the numbers:

• \( 3! = 3 \times 2 \times 1 = 6 \)
• \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)
• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)

As \( n \) increases, \( n! \) becomes extraordinarily large. In our sequence \( a_n = \frac{(-4)^n}{n!} \), the factorial in the denominator indicates that it grows much faster than the exponential growth in the numerator, \((-4)^n\), contributing to the sequence converging towards zero.

Exponential growth is characterized by a quantity growing at a consistent relative rate over equal time periods. In mathematical terms, an exponential function has the form \( b^n \), where \( b \) is a constant base and \( n \) is the exponent. This type of growth can be faster than linear growth but is generally slower than factorial growth for large \( n \).
Consider the sequence given as \( a_n = \frac{(-4)^n}{n!} \). The \((-4)^n\) part represents exponential growth:

• For \( n = 1 \), \( (-4)^1 = -4 \)
• For \( n = 2 \), \( (-4)^2 = 16 \)
• For \( n = 3 \), \( (-4)^3 = -64 \)

The terms grow larger in magnitude with every increase in \( n \). However, when paired with factorial growth in \( n! \), the factorial's rate of increase surpasses the exponential increase by a considerable measure. This relationship leads to the overall sequence \( a_n = \frac{(-4)^n}{n!} \) converging to zero, as the overwhelming growth in the denominator dominates.