Study Notes on Sequences and Series

Fundamental Definitions of Sequences and Series

A sequence is an ordered list of terms denoted by $u_n$, representing the $n$-th term. Common sequences include even numbers, odd numbers, and powers of $2$. A series is the sum of the terms in a sequence. The sum of the first $n$ terms is called a partial sum, denoted as $S_n = u_1 + u_2 + u_3 + ... + u_n$. An infinite series, denoted as $S_{\infty}$, is the sum of all terms in a sequence from $n = 1$ to $\infty$.

The Legend of the Chessboard

The mathematical power of doubling is illustrated by the legend of Sissa ibn Dahir and King Shihram. To teach the king the value of his subjects, Sissa ibn Dahir requested one grain of wheat for the first square of a chessboard, with the amount doubling on every subsequent square. By the $64$th square, the number of grains reaches $2^{63} = 9\,223\,372\,036\,854\,775\,808$, a quantity far exceeding the world's total supply.

Sigma Notation and Properties

Sigma notation ($\sum$) provides a concise way to express the sum of terms. For example, $\sum_{n=1}^{k} u_n$ represents the sum of terms from $n = 1$ up to $k$. Three primary properties apply to sigma notation: the sum of a constant is $\sum_{k=1}^{n} c = nc$, multiplication by a constant allows $\sum c u_k = c \sum u_k$, and the sum of two sequences is $\sum (a_k + b_k) = \sum a_k + \sum b_k$.

General and Recursive Formulas

Sequences can be defined using a general formula, such as $u_n = n^2$, which allows for the direct calculation of any term. Alternatively, they can be defined by a recursive relation, which expresses $u_{n+1}$ in terms of the preceding term $u_n$. A famous example is the Fibonacci sequence, defined in the text as $u_1 = 1, u_2 = 1$ and $u_{n+1} = u_n + u_{n-2}$, resulting in the sequence $1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...$.

Arithmetic Sequences and Series

An arithmetic sequence is characterized by a constant common difference, $d$, between consecutive terms ($u_{n+1} - u_n = d$). The general term is calculated using $u_n = u_1 + (n - 1)d$. For three consecutive terms $a, x, b$, the middle term is the arithmetic mean: $x = \frac{a+b}{2}$. The sum of a finite arithmetic series can be found using $S_n = \frac{n}{2}(u_1 + u_n)$ or $S_n = \frac{n}{2}(2u_1 + (n - 1)d)$. The specific sum of the first $n$ integers is given by $\frac{n(n+1)}{2}$.

Geometric Sequences and Series

A geometric sequence has a constant common ratio, $r$, where $\frac{u_{n+1}}{u_n} = r$. The general term is given by $u_n = u_1 r^{n-1}$. If the terms $a, x, b$ are consecutive, then $x^2 = ab$. The sum of a finite geometric series is calculated as $S_n = \frac{u_1(r^n - 1)}{r - 1}$ or $S_n = \frac{u_1(1 - r^n)}{1 - r}$ for $r \neq 1$.

Infinite Geometric Series and Convergence

An infinite geometric series converges only if the absolute value of the common ratio is less than one (|r| < 1). In this case, the terms approach zero and the sum approaches a finite limit. The formula for the sum of a convergent infinite geometric series is $S_{\infty} = \frac{u_1}{1 - r}$. If |r| > 1, the series diverges and the sum does not exist as a finite number.