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 $L = 1$.

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

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.

$a + (n-1)d$, where $a$ is the first term and $d$ is the common difference.

$a * r^{n-1}$, where $a$ is the first term and $r$ is the common ratio.

$\sum a_n(x-c)^n$

$\sum_{n=0}^{\infty} \frac{x^n}{n!}$

$\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$

$\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}$

$|R_n(x)| \leq \frac{M}{(n+1)!}|x-c|^{n+1}$, where M is the maximum value of the (n+1)th derivative.

$|Error| \leq |a_{n+1}|$, where $a_{n+1}$ is the (n+1)th term of the series.