I'm trying to determine if the following series converges or diverges.
$$\sum_{k=1}^\infty\frac{\sqrt{1+k}}{k}-\frac{1}{\sqrt{k}}$$
Now, I've tried the obvious things, such as the nth term test, trying to write the expression into a single fraction, then apply a comparison test with $\frac{1}{k}$ to get rid of the denominator, however upon taking the limit, it then yeilds zero, which means the test is inconclusive. Furthermore, I don't think the expression is easily integrable, nor can we apply some Taylor or Maclaurin series.
Ideas?