Sums and Products (Sommes et Produits)

A complete set of vocabulary flashcards covering summation formulas, geometric series, index manipulation, double sums, factorials, and finite product rules.

Sum of a Constant Sequence

For any constant $a \in \mathbb{R}$ and integers $p \le n$, the sum of constant terms equals $\sum_{k=p}^n a = (n - p + 1)a$.

Sum of the First $n$ Integers

The closed-form formula for the sum of the first $n$ positive integers: $\sum_{k=1}^n k = \frac{n(n+1)}{2}$.

Sum of the Squares of the First $n$ Integers

The closed-form formula for the sum of the first $n$ squares: $\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$.

Sum of the Cubes of the First $n$ Integers

The closed-form formula for the sum of the first $n$ cubes: $\sum_{k=1}^n k^3 = \frac{n^2(n+1)^2}{4}$.

Geometric Sum from Index 0

For all $n \in \mathbb{N}$, the geometric sum equals $\sum_{k=0}^n q^k = n+1$ if $q = 1$, and $\sum_{k=0}^n q^k = \frac{1-q^{n+1}}{1-q}$ if $q \neq 1$.

General Geometric Sum

For any $q \neq 1$ and $n \ge p$, the sum of geometric terms starting from index $p$ is $\sum_{k=p}^n q^k = q^p \frac{1-q^{n-p+1}}{1-q}$.

Distributivity of Summation

For any constant $\lambda \in \mathbb{R}$ that does not depend on the index $k$, $\sum_{k=1}^n \lambda x_k = \lambda \sum_{k=1}^n x_k$.

Associativity and Commutativity of Summation

The rule that allows splitting the sum of two combined terms into separate sums: $\sum_{k=1}^n (x_k + y_k) = \sum_{k=1}^n x_k + \sum_{k=1}^n y_k$.

Index Shift by Translation ($i = k+m$)

For natural numbers $p \le n$ and integer $m$, translating the summation index by $i = k + m$ gives $\sum_{k=p}^n x_{k+m} = \sum_{i=p+m}^{n+m} x_i$.

Index Shift by Reversal ($i = m-k$)

For natural numbers $p \le n$ and integer $m \ge n$, reflecting the summation index by $i = m - k$ gives $\sum_{k=p}^n x_{m-k} = \sum_{i=m-n}^{m-p} x_i$.

Additive Telescoping Sum

For any sequence of real numbers $x_0, \dots, x_{n+1}$, consecutive terms cancel to give $\sum_{k=0}^n (x_{k+1} - x_k) = x_{n+1} - x_0$.

Double Sum Interchange (Independent Indices)

When the ranges of indices are independent, the summation order can be swapped freely: $\sum_{(i,j) \in [\![1,n]\!] \times [\![1,m]\!]} x_{i,j} = \sum_{i=1}^n \sum_{j=1}^m x_{i,j} = \sum_{j=1}^m \sum_{i=1}^n x_{i,j}$.

Double Sum Interchange ($1 \le i \le j \le n$)

For dependent triangular indices with non-strict inequality: $\sum_{1 \le i \le j \le n} x_{i,j} = \sum_{i=1}^n \sum_{j=i}^n x_{i,j} = \sum_{j=1}^n \sum_{i=1}^j x_{i,j}$.

Double Sum Interchange ($1 \le i < j \le n$)

For dependent triangular indices with strict inequality: $\sum_{1 \le i < j \le n} x_{i,j} = \sum_{i=1}^{n-1} \sum_{j=i+1}^n x_{i,j} = \sum_{j=2}^n \sum_{i=1}^{j-1} x_{i,j}$.

Link Between Single and Double Sums

For real numbers $x_1, \dots, x_n$ with $n \in \mathbb{N}^*$, the expanded square of a sum is given by $\left(\sum_{k=1}^n x_k\right)^2 = \sum_{k=1}^n x_k^2 + 2 \sum_{1 \le i < j \le n} x_i x_j$.

Factorial ($n!$)

For any natural integer $n$, the integer defined by $0! = 1$ and for $n \in \mathbb{N}^*$, $n! = \prod_{k=1}^n k = 1 \times 2 \times \dots \times n$, which satisfies $n! = n \times (n-1)!$.

Product of a Constant Sequence

For all $n \in \mathbb{N}^*$ and constant value $\lambda$, the product of constant terms is $\prod_{k=1}^n \lambda = \lambda^n$.

Multiplicativity of the Product Operator

For all $n \in \mathbb{N}^*$, the product of term-by-term sequence products separates as $\prod_{k=1}^n (x_k y_k) = \left(\prod_{k=1}^n x_k\right)\left(\prod_{k=1}^n y_k\right)$.

Product of a Quotient

When no $y_k$ is zero, the product of quotients factors as $\prod_{k=1}^n \frac{x_k}{y_k} = \frac{\prod_{k=1}^n x_k}{\prod_{k=1}^n y_k}$.

Multiplicative Telescoping Product

When no $x_k$ is zero, for all $n \in \mathbb{N}$, consecutive quotients cancel to yield $\prod_{k=0}^n \frac{x_{k+1}}{x_k} = \frac{x_{n+1}}{x_0}$.