# Technical question regarding showing sqrt(n+1) - sqrt(n) converges to 0

1. Oct 1, 2009

### gmn

1. The problem statement, all variables and given/known data

show the sequence sn= (n+1)1/2 - n1/2 converges to zero

2. Relevant equations

3. The attempt at a solution

I don't have that much of a problem showing the limit goes to zero, rationalize the numerator (or whatever it's called) to get (n+1)1/2 - n1/2 = 1/((n+1)1/2 + n1/2). My question is that I show this goes to zero because sn<1/n(1/2) which goes to zero, but my professor provides a solution where he writes sn<1/2(n1/2). I don't understand why the 2 is there. Is saying that sn<1/(n1/2) insufficient or not true?

Thanks

2. Oct 1, 2009

### lanedance

both are true, and sufficient to show it converges to zero, the 2nd is just a little tighter

$$(n+1)^{1/2} > n^{1/2}$$
then
$$\frac{1}{s_n} = n^{1/2} + (n+1)^{1/2} > n^{1/2} + n^{1/2} = 2n^{1/2}$$
then inverting
$$s_n = \frac{1}{n^{1/2} + (n+1)^{1/2}} < \frac{1}{n^{1/2} + n^{1/2}} = \frac{1}{2n^{1/2}}$$

3. Oct 1, 2009

### dalle

$$\sqrt{n+1}+\sqrt{n}> 2 \sqrt{n}$$.
taking the inverse on both sides yields
$$\frac{1}{\sqrt{n+1}+\sqrt{n}} < \frac{1}{2 \sqrt{n}}$$
your professor is just using a smaller upper bound for $$s_n$$. professors like to use bounds that are as small as possible