Number of primes vs harmonic series

Question

Prove that for $p_k < n < p_{k+1}$, where $p_k$ is k-th prime,

$1+ 1/2 + ... 1/n < k + 1$

I am trying to use the estimate,

$\pi(n) > \ln(n) - 1 $ , but cannot get to the required result

Answer 1

Firstly, for any integer $m>1$, $$\frac{1}{m}<\int_{m-1}^{m}\frac{1}{x}dx$$and by summing this inequality from $2$ to $n$,$$\sum_{m=2}^n\frac{1}{m}<\sum_{m=2}^n\int_{m-1}^{m}\frac{1}{x}dx=\int_{1}^{n}\frac{1}{x}dx=\ln n$$and by OP's $\pi(n) > \ln n -1$,$$\sum_{m=1}^n\frac{1}{m}<\pi(n)+2$$and since $\pi(n)=k$,$$\sum_{m=1}^n\frac{1}{m}<k+2$$and it is not enough to conclude the proof. Instead, you can use $\pi(x)>\frac{x}{\ln x}$ for $x>17$ to prove the inequality for $n>17$ and manually calculate for smaller values of $n$.