  We investigate exact and analytic solutions in $f\left( T\right) $ gravity
within the context of a Friedmann--Lema\^{\i}tre--Robertson--Walker background
space with nonzero spatial curvature. For the power law theory $f\left(
T\right) =T^{n}$ we find that the field equations admit an exact solution with
a linear scalar factor for negative and positive spatial curvature. That
Milne-like solution is asymptotic behaviour for the scale factor near the
initial singularity for the model $f\left( T\right) =T+f_{0}T^{n}-2\Lambda$.
The analytic solution for that specific theory is presented in terms of
Painlev\'{e} Series for $n>1$. Moreover, from the\ value of the resonances of
the Painlev\'{e} Series we conclude that the Milne-like solution is always
unstable while for large values of the indepedent parameter, the field
equations provide an expanding universe with a de Sitter expansion of a
positive cosmological constant. Finally, the presence of the cosmological term
$\Lambda$ in the studied $f\left( T\right) $ model plays no role in the general
behavior of the cosmological solution and the universe immerge in a de Sitter
expansion either when the cosmological constant term $\Lambda$ in the $f\left(
T\right) $ model vanishes.
