Question: Consider the power series: $$ \sum\_{n=1}^{\infty} \frac{(x-b)^{n}}{na^{n}}$$ with a, b >0 :

a) for which x is this series absolute convergent,

b) for which x is this series conditionally convergent,

c) for which x is this series divergent.

So far this is what I have using the ratio test:

$$ \frac{a\_{n+1}}{a\_n} = \frac{\frac{(x-b)^{n+1}}{(n+1)a^{n+1}}}{\frac{(x-b)^n}{na^n}} = 
 \frac{(x-b)na^n}{na^{n+1}+a^{n+1}} = \frac{(x-b)}{na+a^{n+1}}$$

I don't know how to solve the rest of the question, can someone help me with this?

Answer: 
The reason why the conditions are only sufficient is that failure OF A GIVEN FUNCTION to be a Lyapunov function for a system does not mean that no other Lyapunov function exists. This is valid for both linear and nonlinear systems.

There exist converse Lyapunov theorems (the original ones due to Massera, if memory serves me right) establishing that, if a system is stable, then a Lyapunov function exists. The proof in the nonlinear case is neither trivial not completely constructive. 

In the case of linear system, there exists a recipe for constructing a Lyapunov function, which consists in solving the Lyapunov matrix equation. Thus if the equation does not provide a Lyapunov function, none exists, and the system is not stable. Apart from existence of the constructive method, the situations for linear and nonlinear systems are quite similar, technicalities aside.
For a linear system it does provide necessary and sufficient conditions for stability, for a general non-linear system it's not the case because failure of a chosen Lyapunov function to satisfy all the conditions for stability does not mean that the corresponding equilibrium point is not stable. It (may) just indicate that the chosen Lyapunov function is not good enough. For example take this dynamical system $$\ddot{x}(t)=-\sin x-a \dot{x}(t)~~,~~a\in \mathbb{R}\_{>0}$$ Which is $$\begin{align}\dot{x}\_1&=x\_2 \\ \dot{x}\_2&=-\sin x\_1-ax\_2 \end{align}$$ Choose two different Lyapunov function $$\mathbb{R} \times \mathbb{R} \ni (x\_1,x\_2) \mapsto V\_1(x\_1,x\_2):=\left( 1-\cos x\_1 \right)+x\_2^2 \in \mathbb{R} \tag{L.1}$$
$$\mathbb{R} \times \mathbb{R} \ni (x\_1,x\_2) \mapsto V\_2(x\_1,x\_2):=\left( 1-\cos x\_1 \right)+\frac{x\_2^2}{2} \in \mathbb{R} \tag{L.2}$$ 
See what you get using these two different Lyapunov functions