Show that the sequence an=5^n / n ! decreases for n ≥5 . Is...

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Key Concepts

Sequence

A sequence is an ordered list of numbers defined by a specific formula or rule. In the context of the problem, the sequence given is defined by a ratio between an exponential function and a factorial, and its behavior, such as whether it is increasing, decreasing, or non-increasing, is analyzed.

Exponential Function

Exponential functions have the form a^n, where a is a constant and n is the index. They exhibit rapid growth for increasing n, but when compared with factorial functions, their growth rate can become relatively slower for large n.

Factorial Function

The factorial function, denoted as n!, is the product of all positive integers up to n and grows extremely fast as n increases. In sequences like a_n = 5^n/n!, the factorial in the denominator eventually dominates the exponential numerator, influencing the sequence's decreasing behavior for large values of n.

Monotonicity

Monotonicity refers to the consistent behavior of a sequence in terms of increasing or decreasing. A sequence is decreasing if each term is less than or equal to the preceding term, and non-increasing if no subsequent term exceeds the previous one. This concept is key to understanding the trend of the sequence described in the problem.

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