What "tools" are available when determining the convergence of a functional series? I am struggling with determining whether a functional series converges, in general.
$$\sum_{j=1}^\infty f_j$$
When determining whether a functional series(FS) is pointwise, we essentially fix $x \in D(f)$ and send j to $\infty$,(There is more to it than that I know, but for brevity) and see what the limit function is, if we can find a $x \in D(f)$ such that the series is divergent then we have neither pointwise/uniform convergence. 
Since we fix $x$ to determine pointwise, can we use the ratio test, root test, etc in this case?
I posted a question on this;
Pointwise convergence of a functional series.
and I think it is correct, now I have this to tackle
$$\sum_{j=1}^\infty (\frac{x}{j})^j;x \in \mathbb{R}$$
I am going to try determine whether this series is pointwise convergent.
First when $x=0$, the series converges to $0$.
If we fix $x \neq 0$, then by the root test;
$$\lim_{j \to \infty} \frac{|x|}{j} = 0 < 1$$
and so we have pointwise convergence.
We have to look for a different method for uniform convergence, since we know that the rate at which the FS converges will be different for different $x$. This is reflected in the definition. ($\exists N>0 s.t \forall x \in D$...)
One method that my prof suggested was to use the Cauchy criterion for series.
BUT
I can not use it as this function is not bounded. How would I proceed? 
 A: If your series does satisfy a uniform version of the Cauchy criterion on the whole real line then for any $\epsilon$ there would be an integer $N$ so that 
$$\left|  \sum_{j=m}^n \left( \frac{x}j \right)^j \right| < \epsilon$$
for all $n\geq m\geq N$ and all $x\in\mathbb{R}$.  As a very special case of this with $\epsilon=1$ and
not trying quite so hard, we would know that there is an integer $N$ so that
$$\left|   \left( \frac{x}N \right)^N\right| < 1$$ for
all $x\in\mathbb{R}$.  
The uniform Cauchy criterion is equivalent to uniform convergence so you can see that you do not have uniform convergence on $\mathbb{R}$ or indeed on any unbounded set.  Now, to complete the problem, show that you have uniform convergence on any bounded set.
Sometimes when the prof suggests a criterion to use, write it out explicitly and stare at it for quite a while.  Also that "criterion" might be used to prove that something happens or to prove that something does not happen.  You have to consider both directions.  When one fails to work, suspect the opposite.  Go back and forth until a moment of insight arises.
