I'm a little confused how the area approaches infinite. If $\lim_{x\to\infty}\ln(x+1)-\ln(x)=0$, then the area increases less and less as x approaches ∞. If area under the curve is also finite for small $x$ values and doesn't increase much quickly, shouldn't the area under the curve also be finite?
It's just odd that the area starts out small and increases less and less but still manages to approach $\infty$. Please explain to me how this can all be true or if I'm just wrong somewhere