If $\lim_{n \to \infty} a_n \neq 0$, then the series $\sum a_n$ diverges.

If $\lim_{n \to \infty} \frac{a_n}{b_n} = c$, where $c$ is finite and positive, then $\sum a_n$ and $\sum b_n$ either both converge or both diverge.

If $0 \leq a_n \leq b_n$ and $\sum b_n$ converges, then $\sum a_n$ converges. If $a_n \geq b_n \geq 0$ and $\sum b_n$ diverges, then $\sum a_n$ diverges.

If $f(x)$ is continuous, positive, and decreasing for $x \geq 1$, then $\sum_{n=1}^{\infty} f(n)$ and $\int_{1}^{\infty} f(x) dx$ either both converge or both diverge.

If $a_n$ is decreasing and $\lim_{n \to \infty} a_n = 0$, then the alternating series $\sum (-1)^n a_n$ converges.

If $\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = L$, then the series converges if $L < 1$, diverges if $L > 1$, and is inconclusive if $L = 1$.

The distance from the center of a power series within which the series converges.

The error in approximating an alternating series is less than or equal to the absolute value of the first omitted term.

The error in approximating a function using a Taylor polynomial is bounded by the Lagrange Error Bound, which depends on the (n+1)th derivative.

A series $\sum a_n$ converges absolutely if $\sum |a_n|$ converges.

A series $\sum a_n$ converges conditionally if $\sum a_n$ converges but $\sum |a_n|$ diverges.

A function whose domain is the set of natural numbers.

The sum of the terms of a sequence.

Terms approach a specific value (the limit).

Terms do not approach a specific value; the sum is infinite.

Sequence with a constant difference between consecutive terms.

Sequence where each term is the previous term multiplied by a constant ratio.

Series where terms are reciprocals of positive integers.

Series of the form $\sum a_n(x-c)^n$ where $a_n$ are constants and $c$ is a constant.

Series where the terms alternate in sign.

Representation of a function as an infinite sum of terms, with each term being a polynomial function of a single variable.

A Taylor series centered at 0.

Direct Comparison: Compares magnitude directly. Limit Comparison: Compares the limit of the ratio of terms. Direct comparison requires establishing an inequality, limit comparison does not.

Taylor: Expansion around any point c. Maclaurin: Expansion around c=0. Maclaurin is a special case of Taylor.

Alternating Series: Applies only to alternating series, uses the next term. Lagrange: Applies to Taylor polynomials, uses the (n+1)th derivative.

Arithmetic: Constant difference between terms. Geometric: Constant ratio between terms. Arithmetic: Linear growth. Geometric: Exponential growth.