cumsum

The term "cumsum" is a common abbreviation for cumulative sum. It refers to a sequence of partial sums of a given sequence of numbers. In simpler terms, it's the running total of a list of numbers.

For a sequence of numbers, let's say $x_1, x_2, x_3, \ldots, x_n$, the cumulative sum sequence, denoted as $y_1, y_2, y_3, \ldots, y_n$, is calculated as follows:

$y_1 = x_1$

$y_2 = x_1 + x_2$

$y_3 = x_1 + x_2 + x_3$

and so on, until

$y_k = \sum_{i=1}^{k} x_i$ for any $k$ from 1 to $n$.

This concept is widely used in various fields, including mathematics, statistics, computer programming, and finance. In programming, many libraries and functions provide a "cumsum" operation for arrays or lists. For example, in Python's NumPy library, `np.cumsum()` calculates the cumulative sum of an array.

The cumulative sum is useful for understanding trends, calculating running totals for financial reports, analyzing data distributions, and in algorithms that require accumulating values over time or across a dataset. It helps visualize how a sum grows or changes with each added element.