# How do you determine whether the sequence a_n=n!-10^n converges, if so how do you find the limit?

Jul 19, 2017

the sequence $\left\{{a}_{n}\right\}$ diverges

#### Explanation:

We have a sequence defined by:

 a_n = n! -10^n

Our first observation is that for large $n$ then n! grows much faster than any exponential so our intuition tells us that the sequence $\left\{{a}_{n}\right\}$ diverges.

We can demonstrate this using Stirling's Approximation, which states that for large $n$ then:

 n! ~ sqrt(2pin)(n/e)^n

From which we get approximation for ${a}_{n}$ given by:

${a}_{n} = \sqrt{2 \pi n} {\left(\frac{n}{e}\right)}^{n} - {10}^{n}$

And clearly, we have for the dominant term, that:

${n}^{n} \text{ >> } {10}^{n}$