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Converges if the absolute value of the common ratio $|r| < 1$; diverges otherwise.

Converges if $p > 1$; diverges for $p \leq 1$.

Equivalent to the ratio test; converges if $\lim_{{n \to \infty}} \frac{|a_{n+1}|}{|a_n|} < 1$; otherwise, diverges or is inconclusive.

Equivalent to the root test; converges if $\lim_{{n \to \infty}} |a_n|^\frac{1}{n} < 1$; otherwise, diverges or is inconclusive.

A sequence converges if for all $\epsilon > 0$, there exists an $N$ such that for all $m, n \geq N$, $|a_m - a_n| < \epsilon$.

Identical to the alternating series test; converges if the absolute value of the terms decreases monotonically to $0$.

Converges if $\lim_{{n \to \infty}} n\left(\frac{a_n}{a_{n+1}} - 1\right) > 1$; diverges if it is $< 1$; inconclusive if it equals $1$.

Converges if there is a positive number $M$ and $p > 1$ such that $\lim_{{n \to \infty}} n^p\left(\frac{a_n}{a_{n+1}} - 1\right) > M$; otherwise, diverges.

Converges if $\lim_{{n \to \infty}} n(\log(n)\left(\frac{a_n}{a_{n+1}} - 1\right) - 1) > 0$; otherwise, diverges.

Converges if there is a positive sequence $c_n$ such that $\lim_{{n \to \infty}} c_n(\frac{a_n}{a_{n+1}} - 1) > 1$; otherwise, diverges.

Converges if $a_n$ is a monotonic decreasing sequence that tends to $0$ and $b_n$ is bounded, with partial sums also bounded.

Converges if $a_n$ is a monotonic convergent sequence and $b_n$ is a bounded sequence.

Converges absolutely if $\lim_{{n \to \infty}} \left| \frac{a_{n+1}}{a_n} \right| < 1$; diverges if the limit is $> 1$ or infinity; test is inconclusive if the limit equals $1$.

Similar to Gauss's test but modified specifically for hypergeometric series; involves complex analysis for determination.

Used primarily for power series solutions to differential equations; convergence depends on the radius of convergence around singular points.

Converges when the product of $n$ times the difference between consecutive terms $a_n$ and $a_{n+1}$ is convergent.

A particular form of the alternating series test, where convergence is determined if the alternating sum of the sequence's terms tends to a limit.

Converges if $na_n \rightarrow 0$ as $n \rightarrow \infty$ for $a_n > 0$ and $a_n \geq a_{n+1}$ for all $n$.

Converges if there is a constant $C > 0$ such that $\frac{a_n}{a_{n+1}} = 1 + \frac{C}{n} + O(\frac{1}{n^2})$.

Converges if $na_n$ is bounded and $a_n$ tends to $0$ as $n$ tends to infinity.

A series converges if for all $\epsilon > 0$, there exist only finitely many natural numbers $n$ such that $na_n > \epsilon$.

Converges if $\lim_{{n \to \infty}} \frac{\log |a_n|}{\log n} > 1$; otherwise, diverges.

Any conditionally convergent series can be rearranged to converge to any number or even diverge.

Converges absolutely if $\lim_{{n\to\infty}} \sqrt[n]{|a_n|} < 1$; diverges if the limit is $> 1$ or infinity; test is inconclusive if the limit equals $1$.

Used to find the radius of convergence $R$ of a power series; $R = \frac{1}{\limsup_{n \to \infty} \sqrt[n]{|a_n|}}$.

Illustrates a series's convergence via a game with prescribed moves leading to an ever-decreasing remainder; not a formal test.

For an alternating series where terms are represented by the areas of successive vertical rectangles under a decreasing function, the series converges.

If $f(n) = a_n$, the series converges if $\int_{1}^{\infty} f(x) dx$ converges; diverges otherwise.

If $0 \leq a_n \leq b_n$ and $\sum b_n$ converges, then $\sum a_n$ also converges; if $\sum a_n$ diverges, then $\sum b_n$ also diverges.

If $\lim_{{n \to \infty}} \frac{a_n}{b_n} = c$ where $c$ is a positive finite number and $\sum b_n$ converges, then $\sum a_n$ also converges; if $\sum b_n$ diverges, then $\sum a_n$ diverges.

Converges if the sequence $a_n$ is decreasing and $\lim_{{n \to \infty}} a_n = 0$.

A series converges absolutely if the sum of the absolute values of its terms converges.