Prove that $\sum_{k=1}^\infty \sqrt{k+1}-\sqrt{k}$ diverges.

Question

I know that this diverges but I don't know how to prove it rigorously. Should I use the comparison theorem? I know that $\sqrt{k+1}-\sqrt{k}>-\sqrt{k}$ so I just need to show that $\sum_{k=1}^\infty -\sqrt{k}$ diverges. But that is where I am stuck.

It is a telescoping sum and you know what the partial sums are. By definition, a series converges/diverges iff its partial sums converges/diverges. — achille hui, Dec 10 '20 at 03:31

score 0 · Answer 1 · answered Dec 10 '20 at 03:38

0

HINT: For each positive integer $N$:

$$\sum_{k=1}^N (\sqrt{k+1}-\sqrt{k}) = \sqrt{N+1}-\sqrt{1}.$$

Why is this so, and can you finish from here.

answered Dec 10 '20 at 03:38

Mike

20,434

Is it that the limit of that goes to infinity as N goes to infinity? How did you find that $\sqrt{N+1}-\sqrt{1}$? – James Anderson Dec 10 '20 at 05:35

Prove that $\sum_{k=1}^\infty \sqrt{k+1}-\sqrt{k}$ diverges.

1 Answers1