Summation Notation and Algebraic Systems

Vocabulary flashcards covering summation notation, series types, infinite series convergence, large operators, empty index rules, and systems of power sums.

Index of summation

The variable (commonly $i$, $j$, or $k$) that steps through all values from the lower bound to the upper bound within a summation.

Empty sum

A summation where the upper bound is strictly less than the lower bound ($b < a$), which evaluates by definition to the additive identity element $0$.

Scope of a summation

The reach of a summation operator, which extends up to the first addition or subtraction symbol not enclosed in parentheses or part of a larger fractional term.

Einstein summation convention

A notational convention in theoretical physics in which the summation sign is omitted entirely when the index spans an obvious range.

Double sum

A summation nested inside another summation, analogous to two nested for loops, summing the innermost expression over all valid index pairs.

Linearity of summation

The mathematical property permitting constant factors to be factored out ($\sum_{i \in S} a x_i = a \sum_{i \in S} x_i$) and compound terms to be split into separate sums ($\sum_{i \in S} (x_i + y_i) = \sum_{i \in S} x_i + \sum_{i \in S} y_i$).

Geometric series

A series characterized by a constant ratio between consecutive terms, with finite sum $\sum_{i=0}^n x^i = \frac{1 - x^{n+1}}{1 - x}$ and infinite sum $\sum_{i=0}^\infty x^i = \frac{1}{1 - x}$ when $|x| < 1$.

Arithmetic series

A series where the difference between adjacent terms is constant, following the foundational identity $\sum_{i=1}^n i = \frac{n(n + 1)}{2}$.

Harmonic series

The series defined by $\sum_{i=1}^n \frac{1}{i}$, denoted as $H_n$ (the $n$-th harmonic number).

Empty product

A product over an empty index set or where the upper bound is less than the lower bound, which by definition evaluates to the multiplicative identity $1$ (ensuring $0! = 1$).

Big AND operator

An aggregate operator denoted $\bigwedge_{x \in S} P(x)$, equivalent to the universal statement $\forall x \in S : P(x)$, which evaluates to $\text{True}$ over an empty set.

Big OR operator

An aggregate operator denoted $\bigvee_{x \in S} P(x)$, equivalent to the existential statement $\exists x \in S : P(x)$, which evaluates to $\text{False}$ over an empty set.

Infinite sum convergence

The property that a sequence of partial sums $s_n$ approaches a finite value $x$ such that for any $\epsilon > 0$, there exists an integer $N$ where $|s_n - x| < \epsilon$ for all $n > N$.

System of power sums

A system of equations involving sums of powers of variables, such as finding the fourth-power sum $a^4 + b^4 + c^4$ from lower power sums $a + b + c = 4$, $a^2 + b^2 + c^2 = 10$, and $a^3 + b^3 + c^3 = 22$.