Find an integer N that satisfies the limit of the sequence: $\lim_{n\rightarrow\infty} \sqrt{n^2 + n} - n$

Question

Compute the limit then find an integer N that satisfies the limit of the sequence: $$\lim_{n\rightarrow\infty} \sqrt{n^2 + n} - n$$

Now I figured out the limit is $$\frac{1}{2}$$

but I am having trouble finding a N to satisfy the definition of the sequence given that $\epsilon = 10^{-6}$

I've reached an expression: $$\left|(n^2+n)^{\frac{1}{2}} - n - \frac{1}{2}\right| < 10^{-6}$$

What would be a good way to simplify this to solve for N in some way?

Thanks for any input.

Answer 1

$$\sqrt{n^2+n}-n=\frac n{\sqrt{n^2+n}+n}=\frac1{\sqrt{1+\dfrac1n}+1}=\frac12-\epsilon,$$

then

$$n=\frac1{\left(\dfrac1{\dfrac12-\epsilon}-1\right)^2-1}=\frac{(1-2\epsilon)^2}{8\epsilon}<\frac1{8\epsilon}$$

and $$N\ge125000$$ works.

Answer 2

We have $$\frac{1}{2}-\frac{1}{1+\sqrt{1+\frac{1}{n}}}<10^{-6},$$ which gives $$n>\frac{1}{\left(\frac{\frac{1}{2}+10^{-6}}{\frac{1}{2}-10^{-6}}\right)^2-1}=124999.5...$$