Try to prove the inequality

Question

My problem:

Show that the inequality is valid for all sufficiently large $n$

$$ \left\lbrace\frac{1}{{1}^{1/3}}+\frac{1}{{3}^{1/3}}+\cdots+ \frac{1}{{(2n+1)}^{1/3}}\right\rbrace>\frac12 $$

Where $\{\cdot \}$ mean fractional part function.

Answer 1

This is false. The infinite series diverges, but the $n$th term goes to $0$. When all the terms are $<.01$ and the sum first exceeds some large integer $N$, the fractional part will be less than $.01$.