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

Let us look at some details.

By rewriting,

Now, let us evaluate the limit.

#lim_{n to infty}a_n=e^{[ln(1+1/n)]/{1/n}} =e^{lim_{n to infty}{ln(1+1/n)]/{1/n}#

by l'Hopital's Rule,

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To determine whether the infinite sequence (a_n = \left(1 + \frac{1}{n}\right)^n) converges or diverges, we can analyze its behavior as (n) approaches infinity. This sequence is a form of the limit of ((1 + \frac{1}{n})^n) as (n) approaches infinity, known as Euler's number, denoted by (e). If (e), approximately equal to 2.71828, is used to approximate the expression, it gives a clearer understanding of the sequence's behavior. As (n) approaches infinity, the sequence (a_n) converges to (e), meaning it has a finite limit, rather than growing infinitely large or oscillating. Therefore, the infinite sequence (a_n = \left(1 + \frac{1}{n}\right)^n) converges to Euler's number, (e).

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