Question

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

Answer 1

the sequence `{a_n}` 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 `{a_n}` 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(2pin)(n/e)^n -10^n`

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

`n^n " >> " 10^n`