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520 If $f$ is a function whose domain is the positive integers from $1$ to $n$, then the elements in the range of $f,$ taken in the given order, form a finite sequence. \[ f(1),f(2),f(3),\ldots,f(n) \] 520 If $f$ is a function whose domain is all positive integers, then the elements in the range of $f,$ taken in the given order, form an infinite sequence. \[ f(1),f(2),f(3),\ldots,f(n),\ldots \] 521 alternating sequence \[ 2,-4,8,-16,\ldots \] 522 sequence, $n\text{th}$ element or general element \[ \begin{array}{ccc} f(1)&=&a_1\\ f(2)&=&a_2\\ f(3)&=&a_3\\ \vdots&&\vdots\\ f(n)&=&a_n\\ \vdots&&\vdots\\ \end{array} \] 522 explicitly defined sequence \[ a_n=\frac{1+n}{n} \] 522 recursively defined sequence \[ a_n= \left\{ \begin{array}{ll} 2n&\quad\text{if }n\text{ odd}\\ 2a_{n-1}&\quad\text{if }n\text{ even} \end{array} \right. \] 524 The indicated sum of the elements of a finite sequence $a_1,a_2,\ldots,a_n$ is called a finite series given in expanded form. \[ a_1+a_2+\cdots+a_n \] 524 The indicated sum of the elements of an infinite sequence $a_1,a_2,\ldots,a_n,\ldots$ is called an infinite series given in expanded form. \[ a_1+a_2+\cdots+a_n+\cdots \] 525 finite series sigma form \[ \sum_{i=1}^{n}a_i \] 525 infinite series sigma form \[ \sum_{i=1}^{\infty}a_i \] 525 Changing between sigma and expanded form. The letter $i$ is called the index of summation, summation variable, or dummy variable. $i, j, k$ are common index variables. The letter $n$ is the number of terms in the finite series. finite: \[ a_1+a_2+\cdots+a_n=\sum_{i=1}^{n}a_i \] infinite: \[ a_1+a_2+\cdots+a_n+\cdots=\sum_{i=1}^{\infty}a_i \] 532 sum of a constant \[ \sum_{i=1}^{n}c=nc \] 536 positive integers. sum \[ \sum_{i=1}^{n}i=\frac{n\left(n+1\right)}{2} \] positive integers. sum of even \[ \sum_{i=1}^{n}2i=n\left(n+1\right) \] positive integers. sum of odd \[ \sum_{i=1}^{n}{(2i-1)}=n^2 \] 536 positive integers. sum of squares \[ \sum_{i=1}^{n}i^2=\frac{n\left(n+1\right)\left(2n+1\right)}{6} \] 536 positive integers. sum of cubes \[ \sum_{i=1}^{n}i^3 =\left[\frac{n\left(n+1\right)}{2}\right]^2 =\left(\sum_{i=1}^{n}i\right)^2 \] 536 summation properties \[ \sum_{i=1}^{n}{ca_i}=c\sum_{i=1}^{n}a_i \] 536 summation properties \[ \sum_{i=1}^{n}\left(a_i+b_i\right) =\sum_{i=1}^{n}a_i+\sum_{i=1}^{n}b_i \] 536 summation properties \[ \sum_{i=1}^{n}\left(a_i-b_i\right) =\sum_{i=1}^{n}a_i-\sum_{i=1}^{n}b_i \] 541 arithmetic sequence. recursive relation $\forall n\in J,\exists d\in\R.$ \[ a_{n+1}=a_n+d \] 541 arithmetic sequence. common difference \[ d=a_{n+1}-a_n \] 543 arithmetic sequence. general element in terms of $a_1$ and $d.$ \[ \{a_n\}=\{a_1+\left(n-1\right)d\}_{n=1}^\infty \] arithmetic sequence. general element in terms of $a_m$ and $d.$ (not in book, needs proof) \[ \{a_n\}=\{a_m+\left(n-m\right)d\}_{n=m}^\infty \] 545 arithmetic series. $n\text{th}$ partial sum in terms of $a_1$ and $a_n,n$ terms. \[ S_n=\frac{n\left(a_1+a_n\right)}{2} \] 547 arithmetic series. $n\text{th}$ partial sum in terms of $a_1$ and $d.$ \[ S_n=\frac{n}{2}\left[2a_1+\left(n-1\right)d\right] \] arithmetic series. $n\text{th}$ partial sum in terms of $a_n$ and $a_m, n-m+1$ terms, $m\lt n.$ (not in book, needs proof) \[ S_n=\frac{\left(n-m+1\right)\left(a_m+a_n\right)}{2} \] arithmetic series. $n\text{th}$ partial sum in terms of $a_1$ and $d, m\lt n.$ (not in book, needs proof) \[ S_n=\frac{n-m+1}{2}\left[2a_m+\left(n-m\right)d\right] \] 551 geometric sequence. recursive relation $\forall n\in J,\exists r\in\mathbb{R}$ \[ a_{n+1}=a_nr \] 551 geometric sequence. common ratio \[ r=\frac{a_{n+1}}{a_n} \] 553 geometric sequence. general element in terms of $a_1$ and $r$ \[ \{a_n\}=\left\{a_1r^{n-1}\right\}_{n=1}^\infty \] 555 geometric series. $n\text{th}$ partial sum in terms of $a_1$ and $r, n$ terms \[ S_n=a_1\frac{1-r^n}{1-r} \] 557 geometric series. $n\text{th}$ partial sum in terms of $a_1$ and $a_n, n$ terms \[ S_n=\frac{a_1-ra_n}{1-r} \] geometric series. general element in terms of $a_m$ and $r, m\lt n.$ (not in book, needs proof) \[ \{a_n\}=\left\{a_mr^{n-m}\right\}_{n=m}^\infty \] 559 total amount $S$ of a simple ordinary annuity immediately after the $N\text{th}$ payment \[ S=\frac{R\left[\left(1+i\right)^N-1\right]}{i} \] 562 geometric series. $n\text{th}$ partial sum, sequence of partial sums \[ S_n=\sum\limits_{i=1}^n a_1r^{i-1} =a_1\frac{1-r^n}{1-r} \] 563 geometric series convergence. If the $n\text{th}$ partial sum of a geometric series approaches a finite limit $S$ as $n$ increases without bound $(n\rightarrow\infty),$ then the infinite series is said to converge to the sum $S,$ written as shown. \[ \sum\limits_{i=1}^\infty a_1r^{i-1} =\lim\limits_{n\rightarrow\infty} S_n =S \] 564 series. convergence where $S_n=\sum_{i=1}^n a_i$ \[ \sum\limits_{i=1}^\infty a_i =\lim\limits_{n\rightarrow\infty} S_n =S \] 565 geometric series. converges to the given sum $S$ if $\abs{r}\lt 1$ and diverges if $\abs{r}\ge1$ \[ S=\sum\limits_{i=1}^\infty a_1r^{i-1} =\frac{a_1}{1-r} \] 572 $n$ factorial. $n\in J$ \[ n!=n\cdot(n-1)\cdot(n-2)\cdot(n-3)\cdots2\cdot1 \] 572 zero factorial \[ 0!=1 \] 572 simplifying factorial expressions \[ n\cdot(n-1)!=n! \] 573 binomial coefficient where $0\le k\le n,k\in\Z$ \[ \binom{n}{k}=\frac{n!}{k!(n-k)!} \] 574 binomial coefficient. fundamental properties where $0\le k\le n,k\in\Z$
  1. \[ \binom{n}{k}=\binom{n}{n-k} \]
  2. \[ \binom{n}{0}=\binom{n}{n} =1=0! \]
576 binomial theorem \[ (A+B)^n=\sum_{i=0}^n \binom{n}{i}A^{n-i}B^i \] 576 binomial theorem. $k\text{th}$ term in the expansion of $(A+B)^n,$ where $(1\lt k\lt n+1)$ \[ \binom{n}{k-1}A^{n-(k-1)}B^{k-1} \] 581 random experiment, sample space, element \[ \] 582 event \[ \] 583 multiplication principle (fundamental principle of counting) \[ \] 585 permutations of $n$ objects \[ _nP_n=n! \] 585 permutations of $n$ objects taken $r$ at a time \[ _nP_r=\frac{n!}{(n-r)!} \] 586 distinguishable permutations where $n_1+n_2+\cdots+n_k=n$ and $n$ objects are made up of $n_1$ objects of one kind, $n_2$ objects of a second kind, $\ldots,$ $n_k$ objects of a $k\text{th}$ kind. \[ \frac{n!}{n_1!n_2!\cdots n_k!} \] 588 combinations of $n$ objects taken $r$ at a time \[ _nC_r=\binom{n}{r} =\frac{_nP_r}{r!} =\frac{n!}{r!(n-r)!} \] 592 probability of an event $E$ of a finite sample space $S$ in which all elements are equally likely to occur. $n(E)$ is the number of elements of $E$ and $n(S)$ is the number of elements of $S.$ \[ P(E)=\frac{n(E)}{n(S)} \] 592 probability properties
  1. \[ 0\le P(E)\le1 \]
  2. \[ P(S)=1 \]
  3. \[ P(\varnothing)=0 \]
595 event union \[ E=A\cup B \] 595 event intersection \[ E=A\cap B \] 596 probability addition rule \[ P(A\cup B)=P(A)+P(B)-P(A\cap B) \] 597 mutually exclusive events $A$ and $B$ if $A\cap B=\varnothing.$ Thus $P(A\cap B)=$$P(\varnothing)=0$ for mutually exclusive events. \[ A\cap B=\varnothing\\ P(A\cap B)=P(\varnothing)=0 \] 597 probability addition rule for mutually exclusive events \[ P(A\cup B)=P(A)+P(B) \]
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Notes
521 graph of a sequence 523

A sequence cannot be determined uniquely from the first few elements.

Example: \[ 2,4,6,\ldots,a_n,\ldots \]

Two possibile $n\text{th}$ terms: \[ a_n=2n \] or \[ a_n= \left\{ \begin{array}{ll} 2n&\quad\text{if }n\text{ odd}\\ 2a_{n-1}&\quad\text{if }n\text{ even} \end{array} \right. \]

523 pattern recognition. To find the $n\text{th}$ element, assume initial index of 1; experiment with other initial indices. $n\text{th}$ element changes. 524 Sequences exist for which it is not possible to find an expression for the $n\text{th}$ element, e.g. the prime numbers. 524 Each term in a series is the same as the corresponding element in its associated sequence. In a sequence, we refer to $a_n$ as the general element, but in a series we refer to it as the general term. 526 Finding $n\text{th}$ term, and limits 528 shift of index, how to 533 mathematical induction, proving formula for finite series 537 extended form of mathematical induction Prove, using the extended PMI, that the sum of interior angles of an $n$-sided convex polygon is $\left(n-2\right)180^\circ$ for all $n\geq3\in J.$ arithmetic sequence. proof by mathematical induction of general element of 558 A simple annuity also known as an ordinary annuity is a sequence of equal periodic payments of $R$ dollars that are made over $N$ equal time intervals at an interest rate per time interval of $i.$ 566 repeating decimals 567 interval of convergence 577 proof of the binomial theorem 587 combinations 595 Venn diagram