📘Intermediate Algebra Unit 12 – Sequences, Series & Binomial Theorem

Key Concepts and Definitions

• Sequences are ordered lists of numbers that follow a specific pattern or rule
• Terms in a sequence are the individual numbers that make up the sequence and are usually denoted by $a_n$, where $n$ represents the position of the term
• The index of a term is its position in the sequence, starting from 1 (e.g., $a_1$, $a_2$, $a_3$, etc.)
• Series are the sum of the terms in a sequence, denoted by $\sum_{n=1}^{\infty} a_n$, where $a_n$ is the $n$-th term of the sequence
  • Partial sums are the sums of a finite number of terms in a series, denoted by $S_n = \sum_{i=1}^{n} a_i$
• Convergence occurs when the sum of an infinite series approaches a finite value as the number of terms increases
• Divergence occurs when the sum of an infinite series does not approach a finite value or grows without bound as the number of terms increases

Types of Sequences

• Arithmetic sequences have a constant difference between consecutive terms, denoted by $d$
  • The general term of an arithmetic sequence is given by $a_n = a_1 + (n-1)d$, where $a_1$ is the first term and $d$ is the common difference
• Geometric sequences have a constant ratio between consecutive terms, denoted by $r$
  • The general term of a geometric sequence is given by $a_n = a_1 \cdot r^{n-1}$, where $a_1$ is the first term and $r$ is the common ratio
• Recursive sequences are defined by a recurrence relation, where each term is expressed in terms of the previous term(s)
  • Example: the Fibonacci sequence, where $a_1 = 1$, $a_2 = 1$, and $a_n = a_{n-1} + a_{n-2}$ for $n \geq 3$
• Explicit sequences have a closed-form formula that directly relates the index $n$ to the $n$-th term $a_n$
  • Example: the sequence of square numbers, where $a_n = n^2$ for $n \geq 1$
• Alternating sequences have terms that alternate in sign (positive and negative)
  • Example: the sequence $(-1)^n$ for $n \geq 1$ produces the terms $-1, 1, -1, 1, ...$

Arithmetic and Geometric Sequences

• Arithmetic sequences have a constant difference $d$ between consecutive terms
  • To find the common difference, subtract any term from the next term: $d = a_{n+1} - a_n$
  • The sum of the first $n$ terms of an arithmetic sequence is given by $S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}[2a_1 + (n-1)d]$
• Geometric sequences have a constant ratio $r$ between consecutive terms
  • To find the common ratio, divide any term by the previous term: $r = \frac{a_{n+1}}{a_n}$
  • The sum of the first $n$ terms of a geometric sequence is given by $S_n = \frac{a_1(1-r^n)}{1-r}$ for $r \neq 1$ and $S_n = na_1$ for $r = 1$
• Arithmetic and geometric sequences can be combined to form more complex sequences
  • Example: the sequence $a_n = 2n - 3^n$ is the difference of an arithmetic sequence ($2n$) and a geometric sequence ($3^n$)

Series and Their Properties

• Arithmetic series are the sum of the terms in an arithmetic sequence
  • The sum of an infinite arithmetic series is undefined because it diverges (grows without bound)
• Geometric series are the sum of the terms in a geometric sequence
  • The sum of an infinite geometric series converges to $\frac{a_1}{1-r}$ if $|r| < 1$ and diverges if $|r| \geq 1$
• The properties of series include:
  • Linearity: if $\sum a_n$ and $\sum b_n$ converge, then $\sum (ca_n + db_n) = c\sum a_n + d\sum b_n$ for constants $c$ and $d$
  • Comparison: if $0 \leq a_n \leq b_n$ for all $n$ and $\sum b_n$ converges, then $\sum a_n$ also converges
  • Limit comparison: if $\lim_{n \to \infty} \frac{a_n}{b_n} = L > 0$, then $\sum a_n$ and $\sum b_n$ either both converge or both diverge
• Telescoping series are series where most terms cancel out when the partial sums are simplified
  • Example: the series $\sum_{n=1}^{\infty} (\frac{1}{n} - \frac{1}{n+1})$ telescopes to $1$

Convergence and Divergence

• Convergence tests help determine whether a series converges or diverges
  • The $n$-th term test states that if $\lim_{n \to \infty} a_n \neq 0$, then $\sum a_n$ diverges
  • The ratio test compares the limit of the ratio of consecutive terms: if $\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| < 1$, the series converges; if the limit is $> 1$, the series diverges
  • The root test compares the limit of the $n$-th root of the absolute value of the terms: if $\lim_{n \to \infty} \sqrt[n]{|a_n|} < 1$, the series converges; if the limit is $> 1$, the series diverges
• Absolute and conditional convergence:
  • A series $\sum a_n$ converges absolutely if $\sum |a_n|$ converges
  • A series $\sum a_n$ converges conditionally if it converges but $\sum |a_n|$ diverges
• The alternating series test states that if $a_n$ decreases monotonically to 0, then the alternating series $\sum (-1)^{n-1} a_n$ converges

The Binomial Theorem

• The binomial theorem expands powers of binomials $(a+b)^n$ into a sum of terms
  • The general form is $(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$, where $\binom{n}{k}$ are the binomial coefficients
• Binomial coefficients $\binom{n}{k}$ represent the number of ways to choose $k$ objects from a set of $n$ objects, disregarding the order
  • Binomial coefficients can be calculated using the formula $\binom{n}{k} = \frac{n!}{k!(n-k)!}$, where $n!$ represents the factorial of $n$
  • Binomial coefficients satisfy the recursive relation $\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}$, known as Pascal's rule
• The binomial theorem has applications in probability, combinatorics, and algebra
  • Example: expanding $(2x-3)^5$ using the binomial theorem results in a polynomial in $x$

Applications and Problem-Solving

• Sequences and series have numerous real-world applications:
  • Modeling population growth or decay using geometric sequences
  • Calculating compound interest using geometric series
  • Analyzing the convergence of infinite series in physics and engineering
• Problem-solving strategies for sequences and series:
  • Identify the type of sequence (arithmetic, geometric, recursive, explicit) and its properties
  • Determine the general term formula or recurrence relation
  • For series, determine if it converges or diverges using appropriate tests
  • Apply the binomial theorem when expanding powers of binomials
• Combinatorial problems often involve binomial coefficients and the binomial theorem
  • Example: finding the number of ways to select a committee of 3 people from a group of 10 using the combination formula $\binom{10}{3}$

Common Pitfalls and Tips

• Be careful when applying the formulas for the general term and sum of arithmetic and geometric sequences
  • Make sure to use the correct values for $a_1$, $d$, $r$, and $n$
• When using the binomial theorem, ensure that the powers of $a$ and $b$ add up to $n$ in each term
• In convergence tests, pay attention to the limit conditions and strict inequalities
  • For the ratio and root tests, the series is inconclusive if the limit equals 1
• Remember that the $n$-th term test is a divergence test; it cannot prove convergence
• When solving problems involving sequences or series, clearly state your assumptions and show your work step-by-step
• Practice regularly with a variety of problems to reinforce your understanding of the concepts and techniques