Is there a formal name for $ S(k,n) = \sum_{p = 1}^{n} k^p$? Is there a formal name for
$ S(k,n) = \sum_{p = 1}^{n} k^p$
I tried to use the Online Encyclopedia of Integer Sequences but it returned 11036 results. Also this formula has two inputs and I don't know if/how to do that with OEIS.
TL;DR
This is for use with generating test cases with a computer program. I used Catalan number for generating the test cases for binary operators and am now expanding the generator to handle unary operators. 
My first attempt to do this was when ever a unary operator was to be added was to add a binary operator with the second argument set to null, but that resulted in duplicate test cases. 
As a check I would like to be able to calculate the number of correct test cases for unary operators using this function; as Catalan number worked correctly for binary operators. 
The reason I seek the name is because once I have the name I can query with it and get more helpful web pages, e.g. querying with catalan number binary tree test cases quickly led to Every Binary Tree There Is
Yes I am aware that I will need to combine this with the outcome of the Catalan number, but I am taking this one step at a time.
Edits
Simplification formula noted by Hans Lundmark :  
$ S(n,k) = \frac{k(k^n-1)}{(k-1)}$  
 A: This is a geometric series, with two pecularities


*

*the exponents start at $1$ instead of $0$,

*the would-be-zero-exponent-term is $1$ instead of some arbitrary constant.
As far as I know, there is no specific name for this case, and geometric series is an accurate expression. (Also sum of a geometric progression; it is unclear whether one or the other has a bias towards the infinite summation.)
All search engines will return loads of entries not because the term is vague, but because the topic raises a lot of interest.

Your way to describe this summation seems to imply that the common ratio $k$ is an integer and that one considers increasing values of $k$. This is not so natural, one usually deals with $$S(a,r,n)=\sum_{i=0}^n ar^i$$ where $a$ and $r$ are abitrary reals. 
A: Maybe the term "sum of the first $n$ numbers of a geomtric progression with initial value $k$ and ratio $k$" is what you are looking for.
A: While there appears to be no formal agreed upon name, OEIS provides a means that can be referenced and includes references to other sources of information.
From OEIS
A228275 
$
\begin{array}{c|c|c|c|c|c|c|}
  & n & 1 &  2 &   3 &   4 &    5 \\ \hline
k &   &   &    &     &     &      \\ \hline
1 &   & 1 &  2 &   3 &   4 &    5 \\ \hline
2 &   & 2 &  6 &  14 &  30 &   62 \\ \hline
3 &   & 3 & 12 &  39 & 120 &  363 \\ \hline
4 &   & 4 & 20 &  84 & 340 & 1364 \\ \hline
5 &   & 5 & 30 & 155 & 780 & 3905 \\ \hline
\end{array}
$
Column 1: A001477
Column 2: A002378
Column 3: A027444
Column 4: A027445
Column 5: A152031 
Row 1:    A001477
Row 2:    A000918
Row 3:    A029858
Row 4:    A080674 

a(n) is the number of steps which are made when generating all n-step
  random walks that begin in a given point P on a two-dimensional square
  lattice. To make one step means to move along one edge on the lattice.
  - Pawel P. Mazur, Mar 10 2005

This is interesting because Catalan number also references lattice walk.
Row 5:     A104891 
