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

##### 1 Answer

Jul 19, 2017

the sequence

#### Explanation:

We have a sequence defined by:

# a_n = n! -10^n #

Our first observation is that for large

We can demonstrate this using Stirling's Approximation, which states that for large

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

From which we get approximation for

# a_n = sqrt(2pin)(n/e)^n -10^n #

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

# n^n " >> " 10^n #