$\text{if} \ \displaystyle{\lim_{n \to \infty}} a_n \not = 0, \sum a_n \ \text{diverges}$

$\frac{x}{\infty} = 0$, where x is any real number.

$\lim_{x \to \infty} arctan(x) = \frac{\pi}{2}$

$\lim_{n \to \infty} f(n)$

$\sum_{n=1}^{\infty} a_n$

$\lim_{n \to \infty} \frac{P(n)}{Q(n)}$, where P(n) and Q(n) are polynomials.

Divide both numerator and denominator by the highest power of $n$ in the denominator.

$\lim_{n \to \infty} \frac{c}{n^k} = 0$, where $c$ is a constant and $k > 0$.

$\sum_{n=1}^{\infty} a_n \ \text{diverges}$

$\lim_{n \to \infty} a_n$

1. Find $\lim_{n \to \infty} \frac{n^2}{n^2+1}$. 2. Simplify: $\lim_{n \to \infty} \frac{1}{1+\frac{1}{n^2}}$. 3. Evaluate: $\frac{1}{1+0} = 1 \neq 0$. 4. Conclude: The series diverges.

1. Convert to limit: $\lim_{n \to \infty} \frac{3n}{4n-1}$. 2. Simplify: $\lim_{n \to \infty} \frac{3}{4-\frac{1}{n}}$. 3. Evaluate: $\frac{3}{4} \neq 0$. 4. Conclude: The series diverges.

1. Limit: $\lim_{n \to \infty} \frac{n}{\sqrt{n^2+1}}$. 2. Simplify: $\lim_{n \to \infty} \frac{1}{\sqrt{1+\frac{1}{n^2}}}$. 3. Evaluate: $\frac{1}{\sqrt{1+0}} = 1 \neq 0$. 4. Conclude: Diverges.

1. Find $\lim_{n \to \infty} \frac{n^2+1}{5n^2-3}$. 2. Simplify: $\lim_{n \to \infty} \frac{1+\frac{1}{n^2}}{5-\frac{3}{n^2}}$. 3. Evaluate: $\frac{1}{5} \neq 0$. 4. Conclude: The series diverges.

1. Convert to limit: $\lim_{n \to \infty} \frac{2n-1}{n+1}$. 2. Simplify: $\lim_{n \to \infty} \frac{2-\frac{1}{n}}{1+\frac{1}{n}}$. 3. Evaluate: $\frac{2}{1} = 2 \neq 0$. 4. Conclude: The series diverges.

1. Limit: $\lim_{n \to \infty} \frac{n^3}{n^3+2}$. 2. Simplify: $\lim_{n \to \infty} \frac{1}{1+\frac{2}{n^3}}$. 3. Evaluate: $\frac{1}{1+0} = 1 \neq 0$. 4. Conclude: Diverges.

1. Find $\lim_{n \to \infty} \frac{4n}{2n-3}$. 2. Simplify: $\lim_{n \to \infty} \frac{4}{2-\frac{3}{n}}$. 3. Evaluate: $\frac{4}{2} = 2 \neq 0$. 4. Conclude: The series diverges.

1. Convert to limit: $\lim_{n \to \infty} \frac{5n+1}{n-2}$. 2. Simplify: $\lim_{n \to \infty} \frac{5+\frac{1}{n}}{1-\frac{2}{n}}$. 3. Evaluate: $\frac{5}{1} = 5 \neq 0$. 4. Conclude: The series diverges.

1. Limit: $\lim_{n \to \infty} \frac{n^4}{2n^4-1}$. 2. Simplify: $\lim_{n \to \infty} \frac{1}{2-\frac{1}{n^4}}$. 3. Evaluate: $\frac{1}{2} \neq 0$. 4. Conclude: Diverges.

1. Find $\lim_{n \to \infty} \frac{3n-2}{4n+5}$. 2. Simplify: $\lim_{n \to \infty} \frac{3-\frac{2}{n}}{4+\frac{5}{n}}$. 3. Evaluate: $\frac{3}{4} \neq 0$. 4. Conclude: The series diverges.

nth Term Test: Checks if $\lim_{n \to \infty} a_n = 0$. Ratio Test: Uses $\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}|$ to determine convergence/divergence.

nth Term Test: For divergence only. Integral Test: Compares series to an improper integral for convergence/divergence.

nth Term Test: Checks the limit of individual terms. Direct Comparison Test: Compares the series to another known series.

nth Term Test: Focuses on the limit of the nth term. Limit Comparison Test: Compares the limit of the ratio of two series.

nth Term Test: Checks for divergence. Alternating Series Test: Checks for convergence of alternating series.

nth Term Test: For divergence. Root Test: Uses $\lim_{n \to \infty} \sqrt[n]{|a_n|}$ to determine convergence/divergence.

nth Term Test: Checks if the limit of the terms is zero. Geometric Series Test: Checks if the series is a geometric series and |r| < 1 for convergence.

nth Term Test: Checks for divergence based on the limit of the terms. P-Series Test: Checks for convergence/divergence of a series of the form $\sum \frac{1}{n^p}$.

nth Term Test: Checks for divergence. Absolute Convergence Test: Checks if a series converges absolutely by testing the convergence of the absolute values of its terms.

nth Term Test: Checks for divergence. Conditional Convergence Test: Checks if a series converges conditionally, meaning it converges but does not converge absolutely.