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The problem

Does $$\int_{10}^{\infty}\frac{x+ ln x}{x^2} \ dx$$

converge or not?

My attempt

I went ahead and tried to find functions which are strictly larger or smaller than the integrand to see if these converge/diverge but I can only find smaller functions that converge and larger functions that diverge, which tells me nothing about this integral.

The ones I tried are:

$$ \frac{1}{x^2} < \frac{x+ ln x}{x^2} < \frac{x+ x}{x^2} = \frac{2}{x}$$

where the leftmost one converges and the rightmost one diverges

What I need help with

How am I supposed to find good functions to compare to? I don't only have an issue with this specific problem, but with every one of this type. I just happen to find the right function sometimes.

1 Answers 1

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Since $\ln(x)\ge0\forall x\ge1$,

$$\frac{x+\ln(x)}{x^2}\ge\frac{x+0}{x^2}=\frac1x$$

Thus, we know it diverges by the p-series.

  • 0
    Well, that was easy :) Would you like to just write the conclusion that 1/x diverges so I can accept this as an answer? – 2017-02-18
  • 2
    @ejbs No problem :-) Sure hard to find simple things when you see a lot of logarithms and stuff. Took me five minutes to realize $\ln(x)>0$ XD Oh dear... – 2017-02-18