I am interested in the isolated singularity defined over $\mathbb{C}$ by
$$
x\_1^2+\cdots + x\_n^2+x\_{n+1}^k=0,
$$
where $n>2$ and $k>2$.


I would like to know whether this singularity is rational, as in the surface case. Any reference is welcome.



Rationality of higher dimensional du Val singularities
The answer is **yes**. 


In the sequel, I will refer to S. Ishii's book *[Introduction to Singularities](http://www.springer.com/gp/book/9784431550808)* ([Is]). We have the following



> 
> **Proposition.** An $n$-dimensional isolated $1$-Gorenstein singularity $(X, \, x)$ is rational if and only if $\kappa\_{\delta}(X, \, x) = - \infty.$
> 
> 
> 


For a discussion, see [Is, Section 8.4]; the invariant $\kappa\_{\delta}$ is introduced in [Is, Section 6.3]. 


Now, given a hypersurface singularity of the form $(\{f=0\}, \, 0)$, where $$f=x\_1^{a\_1}+ \cdots +x\_{n+1}^{a\_{n+1}},$$
the above condition on $\kappa\_{\delta}$ can be translated as follows: $$\kappa\_{\delta}(X, \, x) = - \infty \iff \sum\_{i=1}^{n+1} \frac{1}{a\_i} >1,$$ see again [Is, Section 8.4]. From this, it follows immediately what you want. 


Note that this also implies that your singularity is canonical, since for a $1$-Gorenstein normal singularity being canonical and being rational are equivalent conditions, see [Is, Corollary 6.2.15].


