Solve T(n)=T(√n)+ n using substitution method
As you mention, 
$$
T(n) = n + \sqrt{n} + \sqrt[4]{n} + \cdots,
$$
and the number of terms is at most $\log \log n$. Hence
$$
n \leq T(n) \leq n + \sqrt{n} \log\log n.
$$
This implies that $T(n) \sim n$.