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

the sequence

We have a sequence defined by:

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

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To determine convergence of the sequence (a_n = n! - 10^n), consider the behavior of the terms as (n) approaches infinity. Notice that (10^n) grows much faster than (n!), meaning (10^n) dominates the growth of the sequence. Hence, (a_n) approaches negative infinity as (n) tends to infinity. Therefore, the sequence diverges.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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