Show that there is a real positive solution

Question

Show that there exists a real positive $n$ such that

$(n+1) \left (1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}\cdots\dfrac{1}{n+1} \right ) \geq 5280n$

Answer 1

Hint: $\displaystyle\sum_{k=1}^n\frac1k\simeq\ln n$.

Answer 2

Suppose for the sake of contradiction that there is no real positive $n$ such that

$$(n+1) \left (1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}\cdots\dfrac{1}{n+1} \right ) \geq 5280n.$$

Then we would have, for all $n$,

$$\left (1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}\cdots\dfrac{1}{n+1} \right ) < 5280\frac{n}{n+1}<5280,$$

implying that the infinite harmonic series converges to a sum less that $5280$. If you know that the harmonic series diverges, you have your contradiction.