Summations and Related Topics Practice Flashcards

Vocabulary terms based on the notes regarding summations, products, various mathematical series, and aggregate operators.

Summations

The discrete versions of integrals, expressed as the sum of a sequence $x_a, x_{a+1}, \text{...}, x_b$ and written as \text{\textsum}_{i=a}^b x_i.

Index of summation

The variable (e.g., $i$) that loops through all values from the lower bound to the upper bound within a summation.

Lower bound (lower limit)

The starting value $a$ for the index variable in a summation.

Upper bound (upper limit)

The ending value $b$ for the index variable in a summation.

Empty sum rule

The rule stating that if the upper bound is less than the lower bound ($b < a$), the sum evaluates to $0$.

Scope of a summation

The range across which the summation applies, extending to the first addition or subtraction symbol not enclosed in parentheses or part of a larger term.

Bijection

A mapping used to rewrite a sum over a finite set $S$ as a sum over indices from $0$ to $|S| - 1$.

Einstein summation convention

A notation system used by theoretical physicists where the summation symbol \text{\textsum}_i is omitted entirely in specific types of sums.

Infinite sums

The limit of a series of partial sums $s_n$ as $n$ approaches infinity, which converges to $x$ if for any \text{\textepsilon} > 0, there is an $N$ such that for all $n > N$, |s_n - x| < \text{\textepsilon}.

Double sums

Two nested summations that function like nested for-loops, summing an innermost expression over all pairs of index values.

Carl Friedrich Gauss

A legendary 18th-century mathematician alleged to have invented the trick for summing the sequence $1, 2, \text{...}, n$ by adding two copies of the sequence in opposite directions.

Linearity

The property of summations where constant factors can be pulled out (\text{\textsum} ax_i = a \text{\textsum} x_i) and sums can be split (\text{\textsum} (x_i + y_i) = \text{\textsum} x_i + \text{\textsum} y_i).

Arithmetic series

A series where the difference between adjacent terms is constant, most commonly represented as \text{\textsum}_{i=1}^n i = \frac{n(n+1)}{2}.

Geometric series

A series where the ratio between adjacent terms is constant, defined by the formula \text{\textsum}_{i=0}^n r^i = \frac{1 - r^{n+1}}{1 - r}.

Harmonic series

The sum of the inverses of integers \text{\textsum}_{i=1}^n \frac{1}{i}, denoted as $H_n$ and characterized in the notes as \text{\textTheta}(n \text{ \textlog}(n)).

Big Pi notation (\text{\textprod})

Mathematical notation used to represent the product of a series of values.

Factorial function ($n!$)

The product of the first $n$ positive integers, defined as n! = \text{\textprod}_{i=1}^n i.

Identity element

The value that when applied to an operation with $x$ does not change $x$; it is $0$ for sums and $1$ for products.

Empty product

Defined to have the value $1$, which is the identity element for multiplication.

Big Intersection (\text{\textbigcap})

An aggregate operator applied to a collection of sets; it is undefined over an empty collection because there is no general identity element.