$a + (n-1)d$, 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.

1. Find $\lim_{n \to \infty} a_n$. 2. If the limit is not 0, the series diverges. 3. If the limit is 0, the test is inconclusive.

1. Choose a series $\sum b_n$ to compare with. 2. Find $\lim_{n \to \infty} \frac{a_n}{b_n} = c$. 3. If $c$ is finite and positive, both series converge or diverge together.

1. Find a series to compare. 2. Establish inequality. 3. If larger converges, smaller converges. If smaller diverges, larger diverges.

1. Verify $f(x)$ is continuous, positive, decreasing. 2. Evaluate $\int_{1}^{\infty} f(x) dx$. 3. If the integral converges, the series converges. If the integral diverges, the series diverges.

1. Check if terms decrease in absolute value. 2. Check if $\lim_{n \to \infty} a_n = 0$. 3. If both conditions are met, the series converges.

1. Find $\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = L$. 2. If $L < 1$, the series converges. 3. If $L > 1$, the series diverges. 4. If $L = 1$, the test is inconclusive.

1. Use the Ratio Test. 2. Solve for $|x - c| < R$. 3. $R$ is the radius of convergence.

1. Find the radius of convergence, R. 2. Test the endpoints $c - R$ and $c + R$ for convergence. 3. Write the interval, including or excluding endpoints based on convergence.

1. Use Alternating Series Error Bound: $|Error| \leq |a_{n+1}|$. 2. Find the smallest $n$ such that $|a_{n+1}|$ is less than the specified error. 3. Sum the first $n$ terms.

1. Find derivatives of the function. 2. Evaluate derivatives at the center. 3. Plug into the Taylor series formula: $\sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!}(x-c)^n$.

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.