Sequences and Series

A numerical sequence is an ordered list of numbers which can often be described by a formula or recurrence relation. Each number within the sequence is called a term. The general term within a sequence is called the $n^{th}$ term.

A numerical series is the sum of a range of terms in a sequence. A finite series describes a series which has $n$ terms ($u_1, u_2, u_3, \dots, u_n$, where $u_n$ is the $n^{th}$ term of the sequence), and thus also has a finite sum. An infinite series describes a series which has infinite terms ($u_1, u_2, u_3, \dots$), which can converge to a finite sum or diverge to infinity.

Sigma notation can be useful when describing a series concisely. For example, the sum: \(u_1+u_2+u_3+u_4+u_5+\dots+u_{100}\) can be rewritten using sigma notation as:

\[\sum_{n=1}^{100} u_n\]

In general,

\[\boxed{\sum_{n=a}^{b}{u_n}}\]

represents the sum of all terms from $n=a$ to $n=b$.

Sigma notation also follows certain important algebraic properties:

Addition Property:

\[\boxed{\sum_{n=1}^k{(a_n+b_n)}=\sum_{n=1}^k{a_n}+\sum_{n=1}^k{b_n}}\]

Constant Multiplication Property:

Given $c$ is a constant:

\[\boxed{\sum_{n=1}^k{c \cdot u_n} = c \cdot \sum_{n=1}^k{u_n}}\]

Arithmetic Sequences and Series

An arithmetic sequence (sometimes called an arithmetic progression) is a sequence where each term differs from the previous term by a constant value, called the common difference, typically denoted as $d$.

The general formula for the $n^{th}$ term of an arithmetic sequence is given by:

\[\boxed{u_n=u_1+(n-1)d}\]

where:

• $u_n$ is the $n^{th}$ term,
• $d$ is the common difference,
• $n$ is the position of the term (e.g., for the 5th term, $n=5$).

---

The sum of a finite arithmetic series is given by:

\[\boxed{ S_n = \frac{n}{2}(u_1 + u_n) \quad \text{OR} \quad S_n = \frac{n}{2}\left(2u_1 + (n - 1)d\right) }\]

where $S_n$ is the sum of the first $n$ terms.

Geometric Sequences and Series

A geometric sequence (sometimes called a geometric progression) is a sequence where each term is the product of the previous term and a constant value, called the common ratio, typically denoted as $r$.

The general formula for the $n^{th}$ term of a geometric sequence is given by:

\[\boxed{u_n=u_1r^{n-1}}\]

---

The sum of a finite geometric series is given by:

\[\boxed{ S_n = \frac{u_1(r^n-1)}{r-1} \quad \text{OR} \quad S_n = \frac{u_1(1-r^n)}{1-r} } \quad \text{where} \quad r \neq 1.\]

where $r$ is the common ratio.

If $\lvert r \rvert < 1$, the geometric series converges as $n \to \infty$, resulting in an infinite geometric series with sum:

\[\boxed{S_{\infty}=\frac{u_1}{1-r}} \quad \text{where} \quad \lvert r \rvert < 1.\]