Sequences, Sigma Notation and Factorials

Unit 10 adds up infinitely many numbers. Before that, this page reviews three tools it leans on: sequences and their limits, factorials, and sigma notation.

Sequences

An infinite sequence is a list $a_1, a_2, a_3, \ldots$ with one term for each counting number $n$. Think of it as a function whose inputs are $1, 2, 3, \ldots$ A formula such as $a_n = \frac{n}{n + 1}$ gives any term directly: $a_1 = \frac{1}{2}$, $a_2 = \frac{2}{3}$, $a_{10} = \frac{10}{11}$.

Some sequences start at $n = 0$. The index matters: $\left(-\frac{1}{2}\right)^{n}$ starting at $n = 0$ and $\left(-\frac{1}{2}\right)^{n - 1}$ starting at $n = 1$ are the same list, $1, -\frac{1}{2}, \frac{1}{4}, \ldots$ To find a formula from a few terms, look for a pattern in the numerators, the denominators and the signs separately; a sign that alternates is a factor $(-1)^n$ or $(-1)^{n+1}$.

A sequence in which each term is the one before times the same number $r$ is a geometric sequence: $a, ar, ar^2, \ldots$ Unit 10 uses these constantly.

Limits of sequences

A sequence converges to $L$ if its terms get as close to $L$ as you like and stay there as $n$ grows: $\lim_{n \to \infty} a_n = L$. Otherwise it diverges.

If $a_n = f(n)$ and $\lim_{x \to \infty} f(x) = L$, then $\lim_{n \to \infty} a_n = L$. So the AB tools work:

• Leading terms for rational expressions: $\lim_{n \to \infty} \frac{3n^2 + 1}{2n^2 - n}$$= \frac{3}{2}$.
• L'Hospital's rule on the function: $\frac{\ln x}{x}$ is an $\frac{\infty}{\infty}$ form, and $\frac{1/x}{1} \to 0$, so $\frac{\ln n}{n} \to 0$.
• Growth rates: as $n \to \infty$, $\ln n$ grows slower than any power $n^p$ with $p > 0$, which grows slower than $e^n$ or $b^n$ with $b > 1$, which grows slower than $n!$.
• The squeeze theorem for signs that alternate: $-\frac{1}{n} \le \frac{(-1)^n}{n} \le \frac{1}{n}$, so $\frac{(-1)^n}{n} \to 0$ (the figure shows a cousin, $1 + \frac{(-1)^n}{n} \to 1$).

Factorials

For a positive integer $n$, $n!$$= n(n - 1)(n - 2)\cdots 3 \cdot 2 \cdot 1$, and by agreement $0! = 1$. Factorials grow very fast: $5! = 120$, $10! = 3{,}628{,}800$.

Ratios of factorials simplify by writing the larger one down until it reaches the smaller one:

Later Unit 10 topics (10.8 and 10.11) lean on exactly these simplifications, and dropping one factor of $(2n + 2)!$ is a common slip.

Sigma notation

$\sum_{k=1}^{m} a_k$ means $a_1 + a_2 + \cdots + a_m$. The letter $k$ is a counter; any letter works. Constants factor out and sums split:

$$\sum_{k=1}^{m} c\,a_k = c\sum_{k=1}^{m} a_k, \qquad \sum_{k=1}^{m} \left(a_k + b_k\right) = \sum_{k=1}^{m} a_k + \sum_{k=1}^{m} b_k.$$

Changing the starting index shifts the formula: $\sum_{k=0}^{3} 2^k$ and $\sum_{k=1}^{4} 2^{k-1}$ are the same sum. In 10.1 the upper limit becomes $\infty$, and the question becomes whether the sum settles to a number at all.