5.1 Sequences

Sequences

A sequence is a function that assigns a value to each positive integer, producing an ordered list of numbers $a_1, a_2, a_3, \ldots$ that may follow a pattern or rule. Understanding sequences is essential because they form the foundation for series, which you'll study throughout the rest of this unit. Before you can determine whether an infinite sum makes sense, you need to understand how individual terms behave as $n$ grows.

General Term Formula Derivation

The general term formula gives you $a_n$ as a function of $n$, so you can compute any term directly without listing all the ones before it.

Arithmetic sequences have a constant difference $d$ between consecutive terms:

$a_n = a_1 + (n - 1)d$

For example, the sequence $3, 7, 11, 15, \ldots$ has $a_1 = 3$ and $d = 4$, so $a_n = 3 + 4(n-1) = 4n - 1$.

Geometric sequences have a constant ratio $r$ between consecutive terms:

$a_n = a_1 \cdot r^{n-1}$

For example, $2, 6, 18, 54, \ldots$ has $a_1 = 2$ and $r = 3$, so $a_n = 2 \cdot 3^{n-1}$.

To find the formula for a given sequence:

1. Look at the terms and identify whether differences or ratios are constant (or whether some other pattern is at work).
2. Use known terms to set up equations and solve for the constants ($a_1$, $d$, or $r$).
3. Plug those constants into the appropriate formula.

Many sequences in Calc II aren't arithmetic or geometric. You might see formulas like $a_n = \frac{n^2}{2n+1}$ or $a_n = \frac{(-1)^n}{n!}$. For these, the goal is to express $a_n$ as an explicit function of $n$ by spotting the pattern across several terms.

Recursive sequences define each term using previous terms rather than giving a closed-form formula. The classic example is the Fibonacci sequence: $F_n = F_{n-1} + F_{n-2}$ with $F_1 = F_2 = 1$. Recursive definitions always need initial conditions to get started.

Sequence Limit Evaluation

The limit of a sequence is the value the terms approach as $n \to \infty$. Formally:

$\lim_{n \to \infty} a_n = L$

means that for every $\varepsilon > 0$, there exists an $N \in \mathbb{N}$ such that $|a_n - L| < \varepsilon$ for all $n \geq N$. In plain terms, no matter how tight a window you put around $L$, eventually all terms of the sequence stay inside that window.

Practical techniques for finding limits:

• Direct substitution / algebraic simplification. Treat $a_n$ like a function of $n$ and evaluate as $n \to \infty$. For rational expressions, divide numerator and denominator by the highest power of $n$. For example:

$\lim_{n \to \infty} \frac{3n^2 + 1}{5n^2 - 2} = \lim_{n \to \infty} \frac{3 + 1/n^2}{5 - 2/n^2} = \frac{3}{5}$

• L'Hôpital's Rule. If the sequence has the form of an indeterminate ratio ($\infty/\infty$ or $0/0$), you can treat $n$ as a continuous variable $x$, apply L'Hôpital's Rule to the corresponding function, and the result gives the sequence's limit.
• Squeeze Theorem. If $b_n \leq a_n \leq c_n$ for all large $n$, and $\lim b_n = \lim c_n = L$, then $\lim a_n = L$. This is especially useful for sequences involving $\sin(n)$ or $\cos(n)$ divided by something growing.

Common limits worth memorizing:

• $\lim_{n \to \infty} \frac{1}{n^p} = 0$ for any $p > 0$
• $\lim_{n \to \infty} r^n = 0$ when $|r| < 1$
• $\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e$
• $\lim_{n \to \infty} \frac{n!}{n^n} = 0$
• $\lim_{n \to \infty} \frac{\ln n}{n} = 0$

When does a limit not exist? If the terms oscillate without settling down (like $a_n = (-1)^n$, which bounces between $-1$ and $1$), the limit does not exist. A sequence that grows without bound ($a_n = n^2$) is said to diverge to infinity.

Convergence vs. Divergence Analysis

A sequence converges if $\lim_{n \to \infty} a_n$ exists and equals a finite number $L$. Otherwise, the sequence diverges. Divergence includes both sequences that grow without bound and sequences that oscillate.

Key techniques for sequences (not series):

• Direct evaluation. Most sequence convergence questions in Calc II come down to computing the limit. If you can find a finite $L$, the sequence converges. If the limit is $\pm \infty$ or doesn't exist, it diverges.
• The Divergence Check. If the terms don't approach zero, the sequence certainly doesn't converge to zero. But be careful: terms approaching zero doesn't guarantee convergence to zero unless you actually verify the limit equals zero. (For example, $a_n = \frac{1}{n}$ does converge to 0, but this reasoning matters more when you get to series.)
• Absolute Value Argument. If $\lim_{n \to \infty} |a_n| = 0$, then $\lim_{n \to \infty} a_n = 0$. This is handy for alternating sequences like $a_n = \frac{(-1)^n}{n}$.
• Monotone Convergence Theorem. If a sequence is both monotone (always increasing or always decreasing) and bounded, then it converges. You don't even need to know the limit to guarantee convergence. This theorem is powerful for recursively defined sequences where computing the limit directly is hard.

Additional Sequence Properties

Monotonicity describes whether a sequence consistently moves in one direction:

• Increasing: $a_n \leq a_{n+1}$ for all $n$ (strictly increasing if $a_n < a_{n+1}$)
• Decreasing: $a_n \geq a_{n+1}$ for all $n$ (strictly decreasing if $a_n > a_{n+1}$)

To test monotonicity, you can examine $a_{n+1} - a_n$ (positive means increasing, negative means decreasing) or look at the ratio $\frac{a_{n+1}}{a_n}$ when all terms are positive (greater than 1 means increasing).

Boundedness means the sequence's values don't escape some fixed interval:

• Bounded above: there exists $M$ such that $a_n \leq M$ for all $n$
• Bounded below: there exists $m$ such that $a_n \geq m$ for all $n$
• Bounded: both conditions hold, so $m \leq a_n \leq M$ for all $n$

Every convergent sequence is bounded (though bounded sequences aren't necessarily convergent; think of $(-1)^n$).

Subsequences are formed by selecting terms from the original sequence while keeping their order. For instance, taking only even-indexed terms $a_2, a_4, a_6, \ldots$ gives a subsequence. The key fact: every subsequence of a convergent sequence converges to the same limit. This works in reverse too: if you can find two subsequences that converge to different values, the original sequence must diverge. (This is exactly how you prove $(-1)^n$ diverges: the even terms converge to 1, the odd terms to $-1$.)

Cauchy sequences capture convergence without needing to know the limit in advance. A sequence is Cauchy if for every $\varepsilon > 0$, there exists $N$ such that $|a_m - a_n| < \varepsilon$ for all $m, n \geq N$. In $\mathbb{R}$, a sequence converges if and only if it is Cauchy. This equivalence is a consequence of the completeness of the real numbers.