How can I find $ a_{n}$ such that $$a_{n} \sim_{n \rightarrow \infty} \sum_{k=1}^n (\ln k)^{1/3} $$ ?
I tried to use integrals:
$$ \int_{k-1}^{k} \ln(t)^{1/3} \mathrm dt\leq \ln(k)^{1/3}\leq \int_{k}^{k+1} \ln(t)^{1/3} \mathrm dt$$ but I cannot compute $$\int_{k-1}^{k} \ln(t)^{1/3} \mathrm dt, \int_{k}^{k+1}\ln(t)^{1/3} \mathrm dt$$
Any idea?