Precalculus Systems, Sequences, and an Introduction to Limits Syllabus

What is a partial fraction decomposition, and when can it be performed?

Rewriting a rational function as a sum of simpler fractions. It requires a proper fraction (degree of numerator less than degree of denominator); otherwise do polynomial division first.

What form does the partial fraction decomposition take for a distinct linear factor $(x-a)$ in the denominator?

A single term $\dfrac{A}{x-a}$ for each distinct linear factor.

What form does the decomposition take for a repeated linear factor $(x-a)^{n}$?

$$\frac{A_{1}}{x-a}+\frac{A_{2}}{(x-a)^{2}}+\cdots+\frac{A_{n}}{(x-a)^{n}}.$$

What form does the decomposition take for an irreducible quadratic factor $(x^{2}+bx+c)$?

A term with a linear numerator: $\dfrac{Ax+B}{x^{2}+bx+c}$. For a repeated irreducible quadratic $(x^{2}+bx+c)^{k}$, include one such term for each power up to $k$.

What is the difference between a sequence and a series?

A sequence is an ordered list of terms $a_{1},a_{2},a_{3},\dots$; a series is the sum of the terms of a sequence, $\sum a_{n}$.

What does summation (sigma) notation $\sum_{i=1}^{n} a_{i}$ mean?

It means $a_{1}+a_{2}+\cdots+a_{n}$, the sum of the terms $a_{i}$ as $i$ runs from the lower limit $1$ to the upper limit $n$.

Define a factorial $n!$ and state the value of $0!$.

$n!=n\cdot(n-1)\cdot(n-2)\cdots 2\cdot 1$, the product of all positive integers up to $n$. By definition $0!=1$.

What is the defining property of an arithmetic sequence, and what is its $n$-th term?

Each term differs from the previous by a constant common difference $d$. The $n$-th term is $a_{n}=a_{1}+(n-1)d$.

State the formula for the sum of the first $n$ terms of an arithmetic series.

$$S_{n}=\frac{n}{2}\left(a_{1}+a_{n}\right)=\frac{n}{2}\left(2a_{1}+(n-1)d\right).$$

What is the defining property of a geometric sequence, and what is its $n$-th term?

Each term is the previous term multiplied by a constant common ratio $r$. The $n$-th term is $a_{n}=a_{1}r^{\,n-1}$.

State the formula for the sum of the first $n$ terms of a geometric series.

$$S_{n}=a_{1}\frac{1-r^{n}}{1-r},\quad r\neq 1.$$

When does an infinite geometric series converge, and what is its sum?

It converges when $|r|<1$, with sum $$S=\frac{a_{1}}{1-r}.$$ If $|r|\geq 1$, the series diverges.

What are the two steps of a proof by mathematical induction?

1) Base case: prove the statement holds for the smallest value (usually $n=1$). 2) Inductive step: assume it holds for $n=k$ (inductive hypothesis) and prove it then holds for $n=k+1$.

What well-known summation formula for $\sum_{i=1}^{n} i$ is often proved by induction?

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

State the Binomial Theorem for $(a+b)^{n}$.

$$(a+b)^{n}=\sum_{k=0}^{n}\binom{n}{k}a^{\,n-k}b^{\,k},$$ where $\binom{n}{k}=\dfrac{n!}{k!(n-k)!}$.

What is the binomial coefficient $\binom{n}{k}$ and how does it relate to Pascal's Triangle?

$\binom{n}{k}=\dfrac{n!}{k!(n-k)!}$ counts combinations; it equals the entry in row $n$, position $k$ of Pascal's Triangle, where each entry is the sum of the two above it.

What is the general (($k+1$)-th) term in the expansion of $(a+b)^{n}$?

$$T_{k+1}=\binom{n}{k}a^{\,n-k}b^{\,k}.$$

State the Fundamental Counting Principle.

If one event can occur in $m$ ways and a second independent event in $n$ ways, then the two together can occur in $m\times n$ ways. This extends to any number of successive events by multiplication.

What is the formula for the number of permutations of $n$ objects taken $r$ at a time?

$$P(n,r)=\frac{n!}{(n-r)!}.$$ Order matters in permutations.

What is the formula for the number of combinations of $n$ objects taken $r$ at a time?

$$C(n,r)=\binom{n}{r}=\frac{n!}{r!(n-r)!}.$$ Order does not matter in combinations.

What is the key distinction between a permutation and a combination?

A permutation counts arrangements where order matters; a combination counts selections where order does not matter. Consequently $P(n,r)=r!\cdot C(n,r)$.

How is the probability of an event $E$ defined for equally likely outcomes?

$$P(E)=\frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}=\frac{n(E)}{n(S)},$$ where $S$ is the sample space. Always $0\leq P(E)\leq 1$.

What is the complement rule in probability?

$$P(E')=1-P(E),$$ where $E'$ is the event that $E$ does not occur.

State the addition rule for the probability of $A$ or $B$.

$$P(A\cup B)=P(A)+P(B)-P(A\cap B).$$ If $A$ and $B$ are mutually exclusive, $P(A\cap B)=0$, so $P(A\cup B)=P(A)+P(B)$.

State the multiplication rule for independent events $A$ and $B$.

$$P(A\cap B)=P(A)\cdot P(B)$$ when $A$ and $B$ are independent.

What does the limit $\lim_{x\to a} f(x)=L$ mean intuitively?

As $x$ gets arbitrarily close to $a$ (from either side, but not equal to $a$), the values of $f(x)$ get arbitrarily close to the single number $L$.

When does the two-sided limit $\lim_{x\to a} f(x)$ exist in terms of one-sided limits?

It exists if and only if both one-sided limits are equal: $$\lim_{x\to a^{-}} f(x)=\lim_{x\to a^{+}} f(x)=L.$$

Does the existence of $\lim_{x\to a} f(x)$ depend on the value $f(a)$?

No. A limit describes the behavior of $f$ near $a$, not at $a$. The limit can exist even if $f(a)$ is undefined or different from the limit value.

How do you estimate a limit numerically?

Build a table of $f(x)$ values for $x$ approaching $a$ from both sides (e.g., $1.9, 1.99, 1.999$ and $2.1, 2.01, 2.001$). If the values converge toward one number $L$ from both sides, that is the estimated limit.

What graphical behavior indicates that a limit does not exist?

The one-sided limits disagree (a jump), the function oscillates without settling, or the values increase/decrease without bound (an infinite discontinuity, e.g., $\lim_{x\to 0}\frac{1}{x^{2}}=\infty$, which fails to exist as a finite value).