Question

In the following problems, find the limit of the given sequence as$n\to \infty$

$\frac{{10}^n}{n!}$

Short answer

The limit of the expression $\frac{{10}^n}{n!}$ is 0

Step-by-step solution

Definition of limits

A limit of a function/expression/sequence is used when the value of the function/sequence cannot be calculated directly or is difficult to calculate at a particular value. It is generally defined as the output obtained when the input is very close to the input value.

Given parameters

The given expression is $\frac{{10}^n}{n!}$and the limit is $n\to \infty$

Ratio Test

Ratio test is a test conducted to find whether any series is converging or diverging.

Solve the limits

Let

Apply the ratio test in the given expression, that is

Solve the obtained limit.

It can be observed that the value ofis 0 which is less than 1, which suggest that the given series is converging and thus the value of limit