# Using the integral test, how do you show whether # (1 + (1/x))^x# diverges or converges?

it converge

By the way you can process like this

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To determine whether the series ((1 + \frac{1}{x})^x) converges or diverges using the integral test, we consider the function (f(x) = (1 + \frac{1}{x})^x).

First, we check the conditions for applying the integral test:

- (f(x)) must be continuous, positive, and decreasing for (x \geq 1).
- (f(x)) must be continuous, positive, and decreasing for (x \geq 1).

Next, we integrate (f(x)) over the interval from 1 to infinity:

[ \int_{1}^{\infty} (1 + \frac{1}{x})^x ,dx ]

If the integral converges, then the series converges; if the integral diverges, then the series diverges.

After integrating, we analyze the result. If the integral converges, it means that the series converges. If the integral diverges, it means that the series diverges.

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