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.

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.