# How do you determine whether the infinite sequence #a_n=e^(1/n)# converges or diverges?

Let us evaluate the limit.

by squeeze the limit into the exponent,

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To determine whether the infinite sequence (a_n = e^{1/n}) converges or diverges, we can analyze its behavior as (n) approaches infinity.

As (n) approaches infinity, the term (1/n) approaches zero. Therefore, (e^{1/n}) approaches (e^0 = 1).

Since the sequence approaches a finite limit of 1 as (n) approaches infinity, the sequence converges.

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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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