(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I'm just curious as to how to think about the following form of equation.

2. Relevant equations

[tex] \int_{3}^{\infty } \frac{1}{x + e^x} \,dx[/tex]

3. The attempt at a solution

What you're trying to do is to test it;

[tex] \frac{1}{x \ + \ e^x} \ < \ \frac{1}{x}[/tex]

[tex] \frac{1}{x} [/tex] diverges

[tex] \frac{1}{x \ + \ e^x} \ < \ \frac{1}{e^x}[/tex]

[tex] \lim_{t \to \infty} \int_{3}^{t} e^{-x}\,dx \ = \ \lim_{t \to \infty} - e^{-x} | \ _3 ^t \ = \ \lim_{t \to \infty} - {\frac{1}{e^t} \ + \ \frac{1}{e^3} [/tex]

so this converges to [tex] \frac{1}{e^3} [/tex].

I don't get how this means the original eq. will also converge?

Both [tex]\frac{1}{x}[/tex] and [tex]\frac{1}{e^x}[/tex] are bigger than the original eq. with one converging and the other diverging.

[tex]\frac{1}{x}[/tex] is bigger than [tex]\frac{1}{e^x}[/tex] but the test of [tex]\frac{1}{x^p}[/tex] is ringing in my ears as a kind of explanation, but I am confused.

Is there an easy way to link all of this together?

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# Homework Help: Comparison Test Question?

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