Sequences and Infinite Series: Foundations and Properties

Sequences

Definition and Representation

A sequence is an ordered list of numbers, typically denoted as {an}, where each term an corresponds to a positive integer n. Sequences can be represented as points on the real line or as points in the plane, where the horizontal axis is the index n and the vertical axis is the value an.

• Infinite sequence: A sequence with infinitely many terms, each having a successor.

• Notation: {an} or {an}∞n=1

Convergence and Divergence of Sequences

A sequence {an} converges to a number L if, for every positive number ε, there exists an integer N such that:

• $|a_n - L| < \varepsilon$ whenever $n > N$

If no such L exists, the sequence diverges.

Divergence to Infinity

A sequence {an} diverges to infinity if, for every number M, there is an integer N such that for all n > N, an > M. Similarly, it diverges to negative infinity if for every number m, there is an integer N such that for all n > N, an < m.

• $\lim_{n \to \infty} a_n = \infty$ or $a_n \to \infty$

• $\lim_{n \to \infty} a_n = -\infty$ or $a_n \to -\infty$

Limit Laws for Sequences

If $\lim_{n \to \infty} a_n = A$ and $\lim_{n \to \infty} b_n = B$, then:

• Sum Rule: $\lim_{n \to \infty} (a_n + b_n) = A + B$

• Difference Rule: $\lim_{n \to \infty} (a_n - b_n) = A - B$

• Constant Multiple Rule: $\lim_{n \to \infty} (k \cdot b_n) = kB$

• Product Rule: $\lim_{n \to \infty} (a_n b_n) = AB$

• Quotient Rule: $\lim_{n \to \infty} \frac{a_n}{b_n} = \frac{A}{B}$ (if $B \neq 0$)

The Sandwich (Squeeze) Theorem for Sequences

If $a_n \leq b_n \leq c_n$ for all $n$ beyond some index $N$, and $\lim_{n \to \infty} a_n = \lim_{n \to \infty} c_n = L$, then $\lim_{n \to \infty} b_n = L$ as well.

Continuous Function Theorem for Sequences

If $a_n \to L$ and $f$ is continuous at $L$ and defined at all $a_n$, then $f(a_n) \to f(L)$.

Limits Involving Functions

If $a_n = f(n)$ for $n \geq n_0$ and $\lim_{x \to \infty} f(x) = L$, then $\lim_{n \to \infty} a_n = L$.

Important Limits of Sequences

• $\lim_{n \to \infty} \frac{\ln n}{n} = 0$

• $\lim_{n \to \infty} \sqrt[n]{n} = 1$

• $\lim_{n \to \infty} x^{1/n} = 1$ (for $x > 0$)

• $\lim_{n \to \infty} x^n = 0$ (for $|x| < 1$)

• $\lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n = e^x$ (for any $x$)

• $\lim_{n \to \infty} \frac{x^n}{n!} = 0$ (for any $x$)

Bounded and Monotonic Sequences

A sequence is bounded from above if there exists $M$ such that $a_n \leq M$ for all $n$, and bounded from below if there exists $m$ such that $a_n \geq m$ for all $n$. If both, it is bounded; otherwise, it is unbounded.

A sequence is nondecreasing if $a_n \leq a_{n+1}$ for all $n$, nonincreasing if $a_n \geq a_{n+1}$ for all $n$, and monotonic if it is either nondecreasing or nonincreasing.

The Monotonic Sequence Theorem

If a sequence is both bounded and monotonic, then it converges.

Infinite Series

Definition and Partial Sums

An infinite series is an expression of the form $a_1 + a_2 + a_3 + \cdots + a_n + \cdots$. The nth partial sum $s_n$ is defined as $s_n = a_1 + a_2 + \cdots + a_n = \sum_{k=1}^n a_k$. The sequence $\{s_n\}$ is the sequence of partial sums.

Convergence and Divergence of Series

If the sequence of partial sums $\{s_n\}$ converges to a limit $L$, the series converges and its sum is $L$:

• $\sum_{n=1}^\infty a_n = L$

If $\{s_n\}$ does not converge, the series diverges.

Examples and Visualizations

Geometric Series

A geometric series with first term $a$ and common ratio $r$ is $a + ar + ar^2 + \cdots$. The series converges if $|r| < 1$ and diverges otherwise:

• $\sum_{n=1}^\infty ar^{n-1} = \frac{a}{1 - r}$ for $|r| < 1$

The nth-Term Test for Divergence

If $\lim_{n \to \infty} a_n$ fails to exist or is not zero, then $\sum a_n$ diverges.

Properties of Series

If $\sum a_n = A$ and $\sum b_n = B$ are convergent series, then:

• Sum Rule: $\sum (a_n + b_n) = \sum a_n + \sum b_n = A + B$

• Difference Rule: $\sum (a_n - b_n) = \sum a_n - \sum b_n = A - B$

• Constant Multiple Rule: $\sum k a_n = k \sum a_n$ (any number $k$)

• Every nonzero constant multiple of a divergent series diverges.

• If $\sum a_n$ converges and $\sum b_n$ diverges, then $\sum (a_n + b_n)$ and $\sum (a_n - b_n)$ both diverge.