Question

Does `a_n=(n + (n/2))^(1/n)` converge?

Answer 1

The sequence is convergent.

Explanation:

`a_n=(n+n/2)^(1/n)=n^(1/n)(3/2)^(1/n)`
so
`lim_(n->oo)(n+n/2)^(1/n)=lim_(n->oo)n^(1/n)lim_(n->oo)(3/2)^(1/n)`

but `lim_(n->oo)n^(1/n)=1` and `lim_(n->oo)(3/2)^(1/n)=1`

so

`lim_(n->oo)a_n=1` and the sequence converges.