Honors Algebra II Unit 9 ReviewSequences and Series

Key Concepts and Definitions

• Sequence represents an ordered list of numbers, where each number is called a term
• Terms in a sequence are usually denoted by $a_n$, where $n$ represents the position of the term
• Common difference $d$ is the constant value added to each term to get the next term in an arithmetic sequence
• Common ratio $r$ is the constant value multiplied to each term to get the next term in a geometric sequence
• Recursive formula defines each term of a sequence using one or more previous terms
  • Example: $a_n = a_{n-1} + d$ for an arithmetic sequence
• Explicit formula allows direct calculation of any term in a sequence based on its position $n$
  • Example: $a_n = a_1 + (n-1)d$ for an arithmetic sequence
• Series is the sum of all terms in a sequence
• Summation notation $\sum$ is used to represent the sum of a series

Types of Sequences

• Arithmetic sequence has a constant difference between consecutive terms
  • Example: 2, 5, 8, 11, 14, ... (common difference is 3)
• Geometric sequence has a constant ratio between consecutive terms
  • Example: 2, 6, 18, 54, 162, ... (common ratio is 3)
• Fibonacci sequence is defined by $F_n = F_{n-1} + F_{n-2}$ with $F_0 = 0$ and $F_1 = 1$
  • Example: 0, 1, 1, 2, 3, 5, 8, 13, ...
• Harmonic sequence is defined by $a_n = \frac{1}{n}$
  • Example: 1, $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$, ...
• Alternating sequence has terms that alternate in sign
  • Example: 1, -2, 3, -4, 5, -6, ...
• Periodic sequence repeats a pattern of terms indefinitely
  • Example: 1, 2, 3, 1, 2, 3, 1, 2, 3, ...

Arithmetic Sequences

• Defined by a common difference $d$ between consecutive terms
• Recursive formula: $a_n = a_{n-1} + d$
• Explicit formula: $a_n = a_1 + (n-1)d$
  • $a_1$ is the first term, $n$ is the position of the term
• To find the common difference, subtract any term from the next term
• Sum of the first $n$ terms of an arithmetic sequence: $S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}[2a_1 + (n-1)d]$
• Arithmetic mean (average) of two numbers $a$ and $b$: $\frac{a+b}{2}$
  • Forms an arithmetic sequence with $a$, the mean, and $b$

Geometric Sequences

• Defined by a common ratio $r$ between consecutive terms
• Recursive formula: $a_n = a_{n-1} \cdot r$
• Explicit formula: $a_n = a_1 \cdot r^{n-1}$
  • $a_1$ is the first term, $n$ is the position of the term
• To find the common ratio, divide any term by the previous term
• Sum of the first $n$ terms of a geometric sequence: $S_n = \frac{a_1(1-r^n)}{1-r}$ for $r \neq 1$
  • If $|r| < 1$, the sum of an infinite geometric series converges to $\frac{a_1}{1-r}$
• Geometric mean of two numbers $a$ and $b$: $\sqrt{ab}$
  • Forms a geometric sequence with $a$, the mean, and $b$

Series and Summation Notation

• Series is the sum of all terms in a sequence
• Summation notation $\sum$ is used to represent the sum of a series
  • Example: $\sum_{i=1}^{n} a_i = a_1 + a_2 + ... + a_n$
• Arithmetic series: $S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}[2a_1 + (n-1)d]$
• Geometric series: $S_n = \frac{a_1(1-r^n)}{1-r}$ for $r \neq 1$
• Infinite series is a series with an infinite number of terms
  • Converges if the sum approaches a finite value as $n \to \infty$
  • Diverges if the sum grows without bound or oscillates
• Partial sum is the sum of the first $n$ terms of a series
  • Denoted by $S_n$

Convergence and Divergence

• Convergent series has a finite sum as the number of terms approaches infinity
  • Example: $\sum_{n=1}^{\infty} \frac{1}{2^n} = 1$ (geometric series with $|r| < 1$)
• Divergent series has a sum that grows without bound or oscillates as the number of terms approaches infinity
  • Example: $\sum_{n=1}^{\infty} \frac{1}{n}$ (harmonic series)
• Geometric series converges if $|r| < 1$ and diverges if $|r| \geq 1$
• $p$-series $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converges for $p > 1$ and diverges for $p \leq 1$
• Alternating series test: if the terms decrease in absolute value and approach 0, the series converges
  • Example: $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$ converges (alternating harmonic series)
• Ratio test: if $\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| < 1$, the series converges; if $> 1$, it diverges

Applications and Problem-Solving

• Modeling growth and decay processes (population, radioactive decay, compound interest)
  • Example: Continuous compound interest formula $A = P \cdot e^{rt}$
• Annuities and installment payments
  • Example: Present value of an annuity $PV = PMT \cdot \frac{1 - (1+r)^{-n}}{r}$
• Fibonacci numbers in nature (spirals in shells, flower petals, pine cones)
• Pascal's triangle and binomial expansions
  • Each number is the sum of the two directly above it
• Geometric series in fractal geometry
  • Example: Cantor set, Koch snowflake
• Zeno's paradoxes and the concept of infinite series
  • Achilles and the tortoise, dichotomy paradox

Advanced Topics and Extensions

• Power series: $\sum_{n=0}^{\infty} a_n(x-c)^n$, used to represent functions as infinite polynomials
  • Example: Taylor series, Maclaurin series
• Fourier series: representing periodic functions as a sum of sines and cosines
  • Applications in signal processing and wave analysis
• Generating functions: a formal power series used to encode the terms of a sequence
  • Example: $G(x) = \sum_{n=0}^{\infty} a_n x^n$
• Recurrence relations and difference equations
  • Example: Fibonacci sequence $F_n = F_{n-1} + F_{n-2}$
• Continued fractions: expressing numbers as a sum of reciprocals
  • Example: Golden ratio $\phi = 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + ...}}}$
• Convergence tests for series (root test, comparison tests, integral test)
  • Determine whether a series converges or diverges
• Applications in physics (quantum mechanics, statistical mechanics)
  • Example: Partition function $Z = \sum_{i} e^{-\beta E_i}$