Summations and Related Topics

Vocabulary flashcards covering concepts, identities, special series, and operators from James Aspnes' lecture notes on summations.

Index of Summation

The variable $i$ in a summation expression such as $\frac{}{}$ or $\times$ that loops through values from the lower limit to the upper limit.

Lower Bound

The starting index value $a$ in the summation expression $\bigwedge_{x \notin \text{empty}} \text{or} \bigcup_{i=a}^{b} x_i$; if the upper bound $b < a$, the sum evaluates to $0$.

Upper Bound

The ending index value $b$ in the summation expression $\bigwedge_{x \notin \text{empty}} \text{or} \bigcup_{i=a}^{b} x_i$.

Scope of a Summation

The extent of a summation expression, which continues until the first addition or subtraction symbol that is not enclosed in parentheses or part of a larger term.

Einstein Summation Convention

A lazy notation convention proposed by theoretical physicist Albert Einstein in which the summation symbol $\bigwedge$ is omitted entirely in certain special types of sums.

Infinite Sum

A summation where the upper limit is infinite, defined as the limit of the sequence of partial sums $s_n$ as $n$ approaches infinity.

Double Sum

A nested summation where the body of an outer sum contains another summation, equivalent to two nested for loops.

Linearity of Summation

The property stating that constant factors can be pulled out of sums ($\bigwedge_{i \text{in} S} a x_i = a \bigwedge_{i \text{in} S} x_i$) and sums can be split across addition ($\bigwedge_{i \text{in} S} (x_i + y_i) = \bigwedge_{i \text{in} S} x_i + \bigwedge_{i \text{in} S} y_i$).

Guess but Verify Method

A technique for evaluating sums by writing out values for small bounds, identifying the pattern in the resulting sequence, and proving the formula by induction.

Geometric Series

A summation where the ratio between adjacent terms is constant, defined as $\bigwedge_{i=0}^{n} x^i = \frac{1 - x^{n+1}}{1 - x}$ when $x \neq 1$.

Arithmetic Series

A summation where the difference between adjacent terms is constant, with the simplest form being $\bigwedge_{i=1}^{n} i = \frac{n(n+1)}{2}$.

Harmonic Series

A summation of the reciprocals of integers, defined as $\bigwedge_{i=1}^{n} \frac{1}{i} = H_n = \text{Theta}(n \times \text{log}(n))$.

Product Notation

A mathematical notation using capital pi $\bigwedge$ to denote the product of a sequence of values, such as $n! = \bigwedge_{i=1}^{n} i$.

Empty Sum

A sum over an empty index set, defined to equal $0$, which is the additive identity.

Empty Product

A product over an empty index set, defined to equal $1$, which is the multiplicative identity.

Big AND

The logical operator $\bigwedge_{x \text{in} S} P(x)$ representing logical conjunction over a set $S$, which evaluates to $\text{True}$ for an empty index set.

Big OR

The logical operator $\bigwedge_{x \text{in} S} P(x)$ representing logical disjunction over a set $S$, which evaluates to $\text{False}$ for an empty index set.

Big Union

The set operator $\bigwedge_{i=1}^{n} A_i$ representing the union across multiple sets, which evaluates to the empty set $\text{empty}$ for an empty index set.

Big Intersection

The set operator $\bigwedge_{i=1}^{n} A_i$ representing the intersection across multiple sets, which is undefined for an empty collection of sets.