Question

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

Answer 1

it converge

Explanation:

By the way you can process like this

`(1+1/x)^x=e^(xln(1+1/x)`

`1/(x+1)<=ln(1+1/x)<=1/x`

`x/(x+1)<=xln(1+1/x)<=1`

`1/(1+1/x)<=xln(1+1/x)<=1`

Take the limit of `1/(1+1/x)` at `-oo` and `oo` for both case it's 1

by the squeeze theorem you can say that `lim x-> oo` `xln(1+1/x) = 1`
and `lim x-> -oo` `xln(1+1/x) = 1`

So `(1+1/x)^x` converge