The set {x ∈ $R^n$: $\sum_{i=1}^n x_i = 1$} I don't really understand the notation of this set: {x ∈ $R^n$: $\sum_{i=1}^n x_i = 1$}
what would be exemplary numerical values for this set?
Would it be e.g. [0.3 ; 0.7] or e.g. [0.1 ; 0.2 ; 0.3 ; 0.4]?
 A: Given a point x in N dimensional space x$_i$ the i-th coordinate.
For example if the point p = (a,b,c), then p$_2$ = b.  
The set itself is a hyperplane passing through the unit of each coordinate.
For n = 2 it is { (x,y) : x + y = 1 }, the line passing (1,0) and (0,1).
For n = 3 it is { (x,y,z) : x + y + z = 1 }, the plane passing through (1,0,0), (0,1,0) and (0,0,1).  
Your examples are correct, your notation isn't.
A: Welcome to MSE! 
First $\Bbb R^n$ is the set of all $n$-dimensional vectors. Using components, an $n$-dimensional vector 
$\mathbf{x}$ is an odrered $n$=tuple $\mathbf{x}=(x_1,x_2,...,x_n)$. 
Or, perhaps it is better to think of this $n$-tuple as the coordinates of a point, as follows: $\mathbf{x}=(x_1,x_2,...,x_n)$ is a point in $n$-dimensional space. 
As pointed out in the comments, if $n=2$ then we are talking about pairs $(x_1,x_2)$. Think of this as a point with coordinates $(x_1,x_2)$, that is a point $\mathbf{x}=(x_1,x_2)$. Then the set 
$\{\mathbf{x}\in\Bbb R^2:\sum_{i=1}^2x_i=1\}$ is the same as the set 
$\{(x_1,x_2):\sum_{i=1}^2x_i=1\}$, that is 
$\{(x_1,x_2):x_1+x_2=1\}$. Usually when $n=2$ one denotes coordinates $(x_1,x_2)$ as $(x,y)$, so 
we could write our set as $\{(x,y):x+y=1\}$. 
This set is the same as the set of all points on the straight line (in the plane) represented by equation 
$x+y=1$, or, what is the same, $y=-x+1$, a line with slope $-1$ and $y$-intersept $1$ (as shown below). 

When $n=3$ then we are talking about triples $(x_1,x_2,x_3)$, that is $\mathbf{x}\in\Bbb R^3$ and $\mathbf{x}=(x_1,x_2,x_3)$. Think of this as a point in (three-dimensional) space with coordinates $(x_1,x_2,x_3)$. Then the set 
$\{\mathbf{x}\in\Bbb R^3:\sum_{i=1}^3x_i=1\}$ is the same as the set 
$\{(x_1,x_2,x_3):\sum_{i=1}^3x_i=1\}$, that is 
$\{(x_1,x_2,x_3):x_1+x_2+x_3=1\}$. Usually when $n=3$ one denotes coordinates $(x_1,x_2,x_3)$ as $(x,y,z)$, so 
we could write our set as $\{(x,y,z):x+y+z=1\}$. 
This set is the same as the set of all points on the plane (in 3D space) represented by equation 
$x+y+z=1$. (Could also be written as $z=1-x-y$.) It is easy to compute the intercepts of this plane with the coordinate axes. Indeed, when $y=z=0$ we get $x=1$, the $x$-intercept. Similarly 
when $x=z=0$ we get $y=1$, the $y$-intercept.
When $x=y=0$ we get $z=1$, the $z$-intercept. 
Here is a plot of this plane (in three dimensional space). 

