I want to use the power series of $(x+1)^{1/2}$ to explain why $$\sum_{n=1}^{\infty}\left({\sqrt{1+{1\over n}}-{1\over n}}\right).$$
converges. I was able to get the expansion of the series using binomial, how can I use that to show that the above series converge?
You can't show that the series converges because it fails the divergence test. That may be why you aren't successful.
$$\lim_{n \to \infty} \left(\sqrt{1+\frac{1}{n}} - \frac{1}{n}\right) = 1 $$
Which implies that the series diverges.