Summations and Related Topics

Flashcards covering the fundamentals of summation notation, formulas, infinite and double sums, and related mathematical operators like products and big logical operators.

Summation

The discrete version of an integral; for a sequence $x_a, x_{a+1}, \text{...}, x_b$, it is written as $\sum_{i=a}^{b} x_i$ and represents the sum of the body for each $i$ from $a$ to $b$.

Sigma

The large jagged symbol ($\sum$) used in summation notation, which is a stretched-out version of a capital Greek letter.

Index of Summation

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

Lower Bound / Lower Limit

The starting value for the index of summation, denoted as $a$ in the notation $\sum_{i=a}^{b} x_i$.

Upper Bound / Upper Limit

The ending value for the index of summation, denoted as $b$ in the notation $\sum_{i=a}^{b} x_i$.

Empty Sum

A summation where the upper bound is less than the lower bound ($b < a$); it evaluates to $0$.

Scope of a Summation

The range of terms included in a sum, which extends to the first addition or subtraction symbol not enclosed in parentheses or part of a larger term like a fraction numerator.

Einstein Summation Convention

A notation style used by theoretical physicists where the $\sum$ symbol is omitted entirely in specific types of sums.

Infinite Sum

The limit of a series $s$ obtained by adding the first term, then the first two terms, the first three, etc.; it converges to $x$ if for any $\epsilon > 0$, there exists an $N$ such that for all $n > N$, $|s_n - x| < \epsilon$.

Double Sum

A summation where the expression inside is another summation, effectively functioning like two nested for-loops.

Simple Arithmetic Series Formula

$\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$.

Geometric Series Formula

$\sum_{i=0}^{n} r^i = \frac{1-r^{n+1}}{1-r}$; this formula works as long as $r \neq 1$.

Harmonic Series

A series of the form $\sum_{i=1}^{n} \frac{1}{i} = H_n$; the transcript identifies this as $\Theta(n \log n)$.

Lineality of Summation

The property allowing constant factors to be pulled out ($\sum a x_i = a ∑ x_i$) and sums inside sums to be split ($\sum (x_i + y_i) = \sum x_i + \sum y_i$).

Guess but Verify Method

A variant of the method for identifying sequences where one writes out the first few values of a sum to recognize a pattern and then proves the formula by induction.

Product Notation

Notation using the capital Greek letter Pi ($\prod$) to multiply a series of values rather than adding them.

Empty Product

A product with an empty index set, which is defined to have the value $1$ (the identity element for multiplication).

Factorial Function

Defined for non-negative $n$ as $n! = \prod_{i=1}^{n} i = 1 \times 2 \times \dots \times n$, where $0! = 1$.

Big Intersection

The aggregate operator $\bigcap_{i=1}^{n} A_i$; it is undefined over an empty collection of sets because there is no identity element.

Big Union Identity

The identity element for the Big Union ($\bigcup$) operator is the empty set.

Big AND Identity

The identity element returned for a Big AND ($\bigwedge$) operation over an empty index set is True.