# How do you determine whether the sequence #a_n=2^n-n^2# converges, if so how do you find the limit?

Using the binomial expansion we have that:

Or:

As:

we have:

So:

and then:

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To determine convergence of the sequence (a_n = 2^n - n^2), consider the behavior of (a_n) as (n) approaches infinity. Note that (2^n) grows exponentially while (n^2) grows polynomially. As (n) becomes large, the exponential term (2^n) dominates the polynomial term (n^2). Therefore, the sequence diverges to positive infinity as (n) approaches infinity.

Hence, the sequence (a_n = 2^n - n^2) diverges, and there is no limit.

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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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