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.

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.

The graph shows that as x approaches infinity, arctan(x) approaches $\frac{\pi}{2}$, which is not zero, indicating divergence.

It suggests the nth term test is inconclusive; the series may converge or diverge, requiring further testing.

If the graph of $a_n$ does not approach the x-axis (y=0) as n goes to infinity, the series diverges.

It indicates that $\lim_{n \to \infty} a_n = 0$, making the nth Term Test inconclusive.

It implies that $\lim_{n \to \infty} a_n$ does not exist or is not equal to zero, indicating divergence.

By showing the trend of the terms as n increases, indicating whether they approach a specific value.

It visually shows whether the terms are approaching a finite value as n increases.

It shows that the function is always between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$, even as x approaches infinity.

The slope indicates the rate of change of the terms; a decreasing slope suggests the terms are getting smaller.

By showing whether the terms stay within a certain range or grow without limit.