Question about convergence of sequence

Question

When $\frac 12\lt m\lt 1$, consider two sequences $a_k=(2\pi k+\frac 1k)^{\frac 1m}$ and $b_k=(2\pi k)^{\frac 1m}$, then $\lim_{k\to \infty}\vert a_k-b_k \vert =0$. But I don't know how this claim proves.

Answer 1

Hint: Note that $$a_{k}-b_k = \frac{\big(1+\frac{1}{2\pi k^2}\big)^{1/m}-1}{\big(\frac{1}{2\pi k}\big)^{1/m}}$$ and then apply L'Hospital as $k\to\infty$.