Precalculus Systems, Sequences, and an Introduction to Limits Flashcards

24 sample cards from the Systems, Sequences, and an Introduction to Limits deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. 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 $0!=1$.

2. 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 $a_{n}=a_{1}+(n-1)d$.

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

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

4. 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 $a_{n}=a_{1}r^{\,n-1}$.

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

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

6. 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.

7. 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 $n=k+1$.

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

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

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

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

10. 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.

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

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

12. 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.

13. 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.

14. 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.

15. 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 $P(n,r)=r!\cdot C(n,r)$.

16. 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 $0\leq P(E)\leq 1$.

17. What is the complement rule in probability?

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

18. 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 $P(A\cup B)=P(A)+P(B)$.

19. 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.

20. 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 $L$.

21. 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: $$\lim_{x\to a^{-}} f(x)=\lim_{x\to a^{+}} f(x)=L.$$

22. 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.

23. 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.

24. 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).

What this deck covers

Systems, Sequences, and an Introduction to Limits flashcards FAQ