Prove that $\displaystyle \lim_{n \to \infty} \ln x = \infty$ using the fact that the harmonic series diverges
Of course, this is obvious graphically, but I have to prove it formally. I based my thinking on this comment:
But I don't understand several things. First of all, why does $\displaystyle\frac{1}{x} > \frac{1}{2}$ imply that the integral is also greater than $1/2$? Secondly, if this is true, how do we use the result that the sum of the $1/x =$ the harmonic series diverges? Is the sum the same as the integral?
Is there a simpler solution to this question?
$$ \ln n=\int_1^n\dfrac{dt}t=\sum_{k=1}^{n-1}\int_{k}^{k+1}\dfrac{dt}t\geq\sum_{k=1}^{n-1}\int_k^{k+1}\dfrac{dt}{k}=\sum_{k=1}^{n-1}\dfrac1{k} $$