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.

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.

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

$\frac{x}{\infty} = 0$, 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 $k > 0$.

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

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