Solvable and generated by finitely many periodic elements not implies periodic

Statement
It is possible to have a group $$G$$ that is a solvable group and is also a group generated by finitely many periodic elements, i.e., $$G$$ has a generating set all of whose elements have finite order, but $$G$$ is not a periodic group, and hence not a periodic solvable group.

This also shows that a solvable group generated by periodic elements need not be periodic.

Proof
Consider the particular example::infinite dihedral group:

$$\langle a,x \mid xax^{-1} = a^{-1}, x = x^{-1} \rangle$$


 * The group is solvable: It has an abelian normal subgroup $$\langle a \rangle$$ with a quotient group that is abelian, isomorphic to cyclic group:Z2.
 * The group is generated by periodic elements: The elements $$x$$ and $$ax$$ form a generating set comprising elements of finite order (both elements have order two).
 * The group is not periodic: However, the element $$a$$ in the group (which generates the cyclic maximal subgroup) has infinite order.