Let M be a positive integer. Show that an=M^n / n !...

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Key Concepts

Ratio Analysis of Successive Terms

Analyzing the ratio between consecutive terms of a sequence is a common method to ascertain monotonic behavior. In this context, by computing the ratio of a_(n+1) to a_n, one can identify conditions under which the ratio is less than one, thereby confirming that the sequence is decreasing for the given range of n.

Factorial Function

The factorial function, denoted as n!, is defined as the product of all positive integers up to n. It grows very rapidly, and its growth rate typically outpaces that of any fixed-base exponential function when n becomes large, making it a fundamental tool in combinatorics and in the study of series convergence.

Exponential Function

An exponential function, such as M^n for a fixed positive integer M, represents a process of repeated multiplication by M. When considered in conjunction with the factorial function, it serves as a basis for comparisons between different types of growth rates in sequences and series.

Monotonicity of Sequences

A sequence is said to be monotonic if it is exclusively non-increasing or non-decreasing. Establishing that a sequence is decreasing involves demonstrating that each term is less than or equal to its predecessor, which can be achieved by comparing the terms directly or via ratios of successive terms.

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