Summations and Related Topics Practice Flashcards

This set of vocabulary flashcards covers the fundamental concepts, notation, definitions, and standard series related to summations and products as described in the lecture notes.

Summation

The discrete version of an integral, written as $\sum_{i=a}^{b} x_i$, where values of a sequence are added from a lower limit to an upper limit.

Index of summation

The variable, such as $i$, $j$, or $k$, that loops through all values from the lower bound to the upper bound in a sum.

Lower limit

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

Upper limit

The ending value for the index of summation, denoted by the variable $b$ in $\sum_{i=a}^{b} x_i$, including both endpoints.

Empty sum

A sum where the upper bound $b$ is less than the lower bound $a$ ($b < a$), resulting in a value of $0$.

Scope of a summation

The extent of the sum which reaches 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 in theoretical physics where the summation symbol $\sum$ is omitted entirely.

Infinite sum

The limit of the series of partial sums $s_n$ as the number of terms approaches infinity.

Double sum

A summation where the internal expression is itself another summation, analogous to two nested for loops.

Arithmetic series

A sum where the difference between adjacent terms is constant, commonly expressed as $\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$.

Geometric series

A sum where the ratio between adjacent terms is constant, calculated as $\sum_{i=0}^{n} r^i = \frac{1-r^{n+1}}{1-r}$, provided $r \neq 1$.

Linearity of summation

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

Guess but verify method

A technique for solving sums by identifying a pattern from the first few values and proving the guessed formula via induction.

Harmonic series

The sum of the reciprocals of integers, denoted as $H_n = \sum_{i=1}^{n} \frac{1}{i}$, which is asymptotically equal to $\Theta(\ln(n))$.

Product notation

The multiplication of a series of values, written using the capital Greek letter pi ($\prod$).

Empty product

A product over an empty index set, which by definition equals the multiplicative identity $1$, leading to the convention that $0! = 1$.

Big AND

The aggregate operator $\bigwedge$ representing universal quantification ($\forall$) over a set, with the identity element True.

Big OR

The aggregate operator $\bigvee$ representing existential quantification ($\exists$) over a set, with the identity element False.

Big Intersection

The aggregate operator $\bigcap_{i=1}^{n} A_i$ used to find the common elements across a collection of sets.

Big Union

The aggregate operator $\bigcup_{i=1}^{n} A_i$ used to combine all elements from a collection of sets, with the identity element being the empty set.