Prove or disprove: $\sum_{k=1}^{\infty}\frac{\sqrt{k+1}-\sqrt{k}}{k\sqrt{k}}$ is convergent
Solution 1:
$$\sum_{k=1}^{\infty}\frac{\sqrt{k+1}-\sqrt{k}}{k\sqrt{k}}< \sum_{k=1}^{\infty}\frac{1}{k\sqrt{k}}$$ $$\sum_{k=1}^{\infty}\frac{1}{k\sqrt{k}}=\sum_{k=1}^{\infty}\frac{1}{k^{\frac{3}{2}}}=\zeta (\frac{3}{2})$$