Defn 1.1. Let $\gamma _{0}(G)=G$, and $\gamma _{c}(G)=[\gamma _{c-1}(G),G]$ for $c\geq 1$. The lower central series of $G$ is a chain of subgroups of $G$: $$G=\gamma_0(G) \geq \gamma_1(G) \geq \cdots \geq \gamma_{c}(G)=1,$$ where the least such $c$ is called the nilpotency class of $G$.
Defn 1.2. Let $G^{(0)}=G$, and $G^{(d)}=[G^{(d-1)},G^{(d-1)}]$ for $d\geq 1$. The derived series of $G$ is a chain of subgroups of $G$: $$G=G^{(0)}\geq G^{(1)}\geq \cdots \geq G^{(d)}=1,$$ where the least such $d$ is called the derived length of $G$.
Defn 2.1. A group $G$ is nilpotent if $\gamma _{c}(G)=1$ for some $c\geq 0$.
Defn 2.2. A group $G$ is soluble(solvable) if $G^{(d)}=1$ for some $d\geq 0$.
It follows from the definitions that $\gamma_1(G)=[G,G]=G^{(1)}$. On the other hand, $\gamma_2(G)=[\gamma_1(G),G]=[[G,G],G]=[\gamma_1(G),G]$ while $G^{(2)}=[G^{(1)},G^{(1)}]=[\gamma_1(G),\gamma_1(G)]$.
My questions are as follows:
We know that $G^{(i)}$ is a subgroup of $\gamma_i(G)$ for all $i$. How can we use it to show that every nilpotent group is solvable?
What are the examples of soluble groups that are not nilpotent?
Thanks in anticipation.
As you say you know $\;G^i\le\gamma_i\;$, if $\;G\;$ is nilpotent there exists $\;c\in\Bbb N\;$ s.t. $\;\gamma_c=1\;$ , and this means $\;G^c\le\gamma_c=1\implies G^c=1\;$ , which means $\;G\;$ is soluble...
OTOH, I think it'd be easier to observe that any central series (say, as the lower central or the upper central series) is always abelian...