TITLE: Is there a name for the sum of increasing powers?
QUESTION [15 upvotes]: I was thinking about the wheat and chessboard story and thinking of the total number of grains of wheat…
$$\sum_{n=0}^{63} 2^n$$
And wondered if there is a name for a sum like this?
$$2^0 + 2^1 +2^2 + 2^3 + \cdots$$

REPLY [4 votes]: As you are looking for novel names, this is also a simple case of the more generic notion of hypergeometric series 
$$\sum_{k}r_k \,,$$
with $r_0 = 1$, and the ratio of two consecutive terms is a rational function, a ratio of two polynomials $P$ and $Q$ in the summation index  $k$
$$ \frac{r_{k+1}}{r_k}= \frac{P(k)}{Q(k)}\,. $$
In your case, you can choose $P$ and $Q$ such that their ratio is equal to $2$. When the ratio is constant, it is called  a geometric series (as answered here). As a reminder, it is a sum of terms in geometric progression (se.math) like $1,r,r^2,r^3,\ldots$, whose name (the geometry part) is illustrated by the following figure:

Hypergeometric series are also connected to chess. A rook is a move on a chessboard. Some  have developed studies  some types of permutations as the placement of a number of rooks on a chessboard-like grid, see for instance Rook theory and hypergeometric series,  J. Haglund, 1996.