Is there a name for the sum of increasing powers?

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?

$$2^0 + 2^1 +2^2 + 2^3 + \cdots$$

Solution 1:

This is called a geometric sum/series according as $N$ is finite or not. More generally, it is of the form $$ \sum_{n=0}^Nar^n$$ where $N$ may be "equal to" infinity. In the case where $N$ is infinite, if the series is to converge, we require $\lvert r\rvert<1$. Furthermore, for $N$ finite $$ \sum_{n=0}^Nar^n=a\times\frac{1-r^{N+1}}{1-r},\:\:\:\: r\ne1$$ and $$ \sum_{n=0}^\infty ar^n=\frac{a}{1-r},\:\:\:\:\lvert r\vert<1.$$

Solution 2:

It's mainly a well known value $2^{n+1}-1$ . Write it in base $2$.

It is also OEIS sequence A000225 :

$2^n - 1.$ (Sometimes called Mersenne numbers, although that name is usually reserved for A001348.)

Solution 3:

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.