Solution verification of $\sum_{n=1}^{\infty}\frac{b^{2n^2}}{n^n}$ I want to study
$$\sum_{n=1}^{\infty}\frac{b^{2n^2}}{n^n}\qquad\text{with $b\in\mathbb{R}$.}$$
I have a positive term series and so I have decide to apply the root test.
In particular:
$$\Bigg(\frac{b^{2n^2}}{n^n}\Bigg)^{\frac{1}{n}}=\Bigg(\frac{b^{2n}}{n}\Bigg)\to\begin{cases} 0& \text{if $-1\leq b\leq 1$}\\ \infty& \text{if $b<-1\vee b>1$.}\end{cases}$$
So if $-1\leq b\leq 1$ then the series converges; otherwise, for $b<-1\vee b>1$, the series diverges.
Question: is my attempt right?
 A: Because I am generally opposed to the use of Math SE as a means for checking solutions to homework problems (the solution-verification tag notwithstanding), I am going to attempt to address more than just the headline question here ("Is my attempt right?") and try to address some other issues of presentation and style, as well (remember, the goal of mathematics is to communicate ideas).  This is meant to address what I see as the the underlying question of all solution-verification questions:  "How can I improve my answer?"
Style and Presentation
There are a couple of places where your presentation could be improved.  While these comments are a matter of opinion, and there are probably folk here who would argue against me, I am going to give my suggestions.  At the very least, these are issues you should consider, even if you don't agree with me.

*

*I generally believe that it is better to use language than notation, unless the idea is so complicated or convoluted that language is going to get in the way.  Hence, for example
\begin{align}
&{\color{red}{✗}} && \text{for $b<-1 \lor b>1$,} \\
&{\color{green}{\checkmark}} && \text{for $b < -1$ or $b > 1$.}
\end{align}


*In this presentation, it might be easier to get rid of the "or" entirely, and use the simplified statements "$|b|<1$" and "$|b|>1$".  Here is an example of where a more concise, notation-heavy exposition might be preferable.


*I like the imperative voice (or even the passive voice) for mathematical writing.  You don't need to explain to the reader what you are going to do, or what we have done.  Just do it.  In this case, you don't need to explain that you are going to apply the root test.  Just tell the reader what to do, and state the result.


*Think carefully about the statements of the theorems you are applying.  The root test is typically stated as something like

Let $(a_n)$ be a sequence and define
$$ \rho = \lim_{n\to \infty} |a_n|^{1/n}.$$

*

*If $\rho < 1$, then $\sum a_n$ converges absolutely,

*if $\rho > 1$, then $\sum a_n$ diverges, and

*if $\rho = 1$, then the root test is inconclusive.


When you apply the root test, make sure that you are working with the correct formulae and that all of the appropriate hypotheses apply.  With respect to your presentation, you mention at the beginning of your argument that all of the terms are positive, and so you drop the absolute values immediately—I think that you can (a) save some words and (b) aid clarity by not doing this.


*Elide tedious computation.  Honestly, you did an excellent job of this, and I have no complaints.  However, one of the things that students will often do is write out every single computational step in excruciating detail, so I want to point out that this really isn't necessary.  Consider your audience, and let them fill in gaps which should be routine.
Mathematics
This depends a little on what the exercise was meant to show, but it seems to me that you have left out an important case:  what happens when the root test is inconclusive?  You should either explain why you don't care about those cases (perhaps all we need to know is the radius of convergence of this series), or work them out explicitly.  That said (as pointed out in the comments below), the root test takes care of the boundaries, as well.
Other than that, I have no complaints.
Rewrite
If I were given this problem as a student, my writeup would likely look something like the following:

Exercise: Determine real values of $b$ such that the series
$$\sum_{n=1}^{\infty} \frac{b^{2n^2}}{n^n} $$
converges.

Solution:  Observe that
$$\lim_{n\to\infty} \left| \frac{b^{2n^2}}{n^n} \right|^{1/n}
= \lim_{n\to\infty} \left( \frac{|b|^{2n}}{n} \right)
\xrightarrow{n\to\infty} \begin{cases}
0 & \text{if $|b|\le 1$, and} \\
\infty & \text{if $|b| > 1.$}
\end{cases}
$$
By the root test, the series converges absolutely when $|b| \le 1$ and diverges otherwise.
