Does 1/n! converge?

Understand the Problem

The question is asking whether the sequence formed by the terms 1/n! converges as n approaches infinity. Convergence in this context refers to whether the terms approach a specific value as n increases indefinitely.

Answer

The sequence converges to 0.

Steps to Solve

Identify the sequence

The terms of the sequence are given by the function $a_n = \frac{1}{n!}$, where $n$ is a positive integer.

Evaluate the factorial

As $n$ increases, the factorial $n!$ grows very quickly. For example,

$$ 1! = 1, \quad 2! = 2, \quad 3! = 6, \quad 4! = 24, \quad 5! = 120, \ldots $$

Determine the behavior of the sequence

Let's analyze the limit of the sequence as $n$ approaches infinity:

$$ \lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{1}{n!} $$

Since $n!$ increases without bound as $n$ goes to infinity, this means $\frac{1}{n!}$ gets smaller and smaller.

Conclude the convergence

Based on the above limit, we conclude:

$$ \lim_{n \to \infty} \frac{1}{n!} = 0 $$

This shows that the sequence converges to 0.

The sequence converges to 0.

More Information

The sequence $\frac{1}{n!}$ is an example of a decreasing sequence that approaches zero as $n$ increases. This is significant in mathematical analysis and shows how quickly factorials grow compared to linear sequences, illustrating the rapid decay of the terms in this series.

Tips

Confusing convergence with divergence: Make sure to differentiate between a sequence that converges to a limit and one that grows indefinitely.
Misunderstanding factorial growth: Remember that factorials grow faster than exponential functions, leading to terms approaching zero.

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