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

Answer 1

it converge

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
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Answer 2

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:

  1. (f(x)) must be continuous, positive, and decreasing for (x \geq 1).
  2. (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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Answer from HIX Tutor

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.

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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