Question

How do you determine whether the sequence `a_n=sqrt(n^2+n)-n` converges, if so how do you find the limit?

Answer 1

See below.

Explanation:

`sqrt(n^2+n)-n=(n^2+n-n^2)/(sqrt(n^2+n)+n) = n/(sqrt(n^2+n)+n)`

but

`n/(sqrt(n^2+n)+n)=n/(n(sqrt(1+1/n)+1))=1/(sqrt(1+1/n)+1)`

so

`lim_(n->oo)a_n=lim_(n->oo)1/(sqrt(1+1/n)+1) = 1/2`

so the sequence is convergent and converges to `1/2`