Limit of natural log

Question

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:

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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?

$f(x)\ge C $ gives $\int_{a}^b f(x) dx \ge (b-a)C$. Here, $b-a$ is 1 for all of your definite integrals. — David P, Mar 04 '14 at 01:47
Ok, would it be correct to sum up the integrals and say that the sum > the sum of c, which we know diverges? — kiwifruit, Mar 04 '14 at 01:48

score 1 · Accepted Answer · answered Mar 04 '14 at 02:04

1

$$ \ln n=\int_1^n\dfrac{dt}t=\sum_{k=1}^{n-1}\int_{k}^{k+1}\dfrac{dt}t\geq\sum_{k=1}^{n-1}\int_k^{k+1}\dfrac{dt}{k}=\sum_{k=1}^{n-1}\dfrac1{k} $$

answered Mar 04 '14 at 02:04

Martin Argerami

205,756

You are amazing... – recursive recursion Mar 04 '14 at 02:07

Limit of natural log

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