If the terms of a series don't approach zero, the series cannot converge, so it must diverge.

If the terms approach zero, the series might converge, but further testing is needed.

Convert the series notation to limit notation: $\lim_{n \to \infty} a_n$.

The sum of the terms in the series approaches a finite value as the number of terms increases.

An incorrect limit evaluation leads to a wrong conclusion about the divergence of the series.

Simplification makes the limit easier to evaluate and reduces the chance of errors.

If the terms don't get smaller and smaller, their sum will grow without bound, preventing convergence.

The angle whose tangent is a given number.

To determine the convergence or divergence of series and integrals.

A sequence is a list of numbers, while a series is the sum of the numbers in a sequence.

If $\lim_{n \to \infty} a_n \neq 0$, then $\sum a_n$ diverges.

The sum of the series does not approach a finite value.

The nth term of the series.

The value that a function or sequence approaches as the input or index approaches some value.

The sum of the terms of a sequence.

The nth Term Test is inconclusive; the series may converge or diverge.

The test does not provide enough information to determine convergence or divergence.

The inverse tangent function, denoted as $arctan(x)$ or $tan^{-1}(x)$.

The value that a function approaches as the input approaches some value.

$n$ represents the index or term number in a sequence or series, approaching infinity.

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.