Verify if the series $\sum a_n$ with $a_n=\sqrt{n+1}-\sqrt{n}$ is
convergent or divergent.
What I did is
$$a_n=(\sqrt{n+1}-\sqrt{n})\times\frac{\sqrt{n+1}+\sqrt{n}}{\sqrt
{n+1}+\sqrt{n}}$$
$$=\frac{1}{\sqrt{n+1}+\sqrt{n}}$$
$$<\frac{1}{2\sqrt{n}}=b_n$$
Since $b_n$ is monotone decreasing and $b_n\rightarrow 0$ when $n\rightarrow \infty$ then $b_n$ is convergent.
Using the comparison test, we have that $0\leq a_n\leq b_n$. If $b_n$ is convergent then $a_n$ is convergent.
Is it right?
Best Answer
The series diverges. This can be seen using the fact that it is a "telescoping series".
The $k^\text{th}$ partial sum can be written as follows:
\begin{align} S_k &= \displaystyle\sum_{n=1}^k \Big(\sqrt{n+1} - \sqrt{n}\Big)\\\\ &= \big(\sqrt{2} - \sqrt{1}\,\big) + \big(\sqrt{3} - \sqrt{2}\,\big) + \big(\sqrt{4} - \sqrt{3}\,\big) + \cdots +\big(\sqrt{k+1} - \sqrt{k}\,\big)\\\\ &=-\sqrt{1} + \big(\sqrt{2} -\sqrt{2}\,\big) + \big(\sqrt{3} -\sqrt{3}\,\big) + \big(\sqrt{4} - \cdots -\sqrt{k}\,\big) + \sqrt{k+1}\\\\ &=\sqrt{k+1} - 1\\ \end{align}
As $k$ goes to infinity, this diverges, so the infinite sum does not converge.
$$\displaystyle\sum_{n=1}^\infty \Big(\sqrt{n+1} - \sqrt{n}\Big) = \lim_{k\to\infty} S_k = \infty$$