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

1 Answer
Steve M
Jul 19, 2017

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 #

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