July 02, 2026 - Calculus 2 - Series Convergence and Calculus Tests Flashcards

Vocabulary practice flashcards covering absolute and conditional convergence, the alternating series test, the ratio test, and the root test based on the lecture notes.

Absolutely Convergent

A series is classified this way if the absolute value of the series converges.

Conditionally Convergent

A series that converges itself, but its absolute value series diverges, such as the alternating harmonic series.

Alternating Series Test Conditions

Two conditions must be met: the limit as $n$ goes to infinity of the sequence must be zero ($\lim_{n \to \text{infinity}} a_n = 0$), and the terms must be getting smaller ($a_{n+1} \text{ less than or equal to } a_n$).

Harmonic Series

The series $\sum \frac{1}{n}$, which is a P-series with $p = 1$ and is divergent.

Alternating Harmonic Series

A famous example of a conditionally convergent series that converges by the alternating series test but whose absolute value diverges.

Limit Comparison Test

A test used to determine convergence by taking the limit of the ratio of the series in question and a known series; a finite non-zero result means both series share the same convergence behavior.

Ratio Test

A test involving the limit of the absolute value of the $(n+1)$ term divided by the $n$th term: $\lim_{n \to \text{infinity}} | \frac{a_{n+1}}{a_n} |$.

Ratio Test (Convergence)

If the limit result of the ratio test is less than $1$, the series converges absolutely.

Ratio Test (Divergence)

If the limit result of the ratio test is greater than $1$ or infinity, the series diverges.

Ratio Test (Inconclusive)

If the limit result of the ratio test equals exactly $1$, the test fails and provide no information about convergence.

Root Test

A test used when terms contain an exponent of $n$, involving the limit of the $n$th root of the absolute value of the series: $\lim_{n \to \text{infinity}} \text{nth} \text{-root}(|a_n|)$.

Factorial ($n!$)

A mathematical operation where a number is multiplied by every number less than it down to $1$; for example, $3! = 3 \times 2 \times 1$.

Factorial Expansion Rule

The property that allows rewriting a factorial as $(n+1)! = (n+1) \times n!$ to simplify ratios.

$0$ Factorial ($0!$)

Defined as being equal to $1$.

Euler's Number ($e$) in Limits

The mathematical constant arising from the specific limit $\lim_{n \to \text{infinity}} (1 + \frac{1}{n})^n$, approximately equal to $2.718$.

P-Series

A series of the form $\sum \frac{1}{n^p}$; it converges if $p > 1$ and diverges if $p \text{ less than or equal to } 1$.

Absolute Convergence Distinction

Absolute convergence implies the series converges regardless of whether it oscillates or has negative terms, whereas conditional convergence depends on the signs/alternation.

Leading Coefficient Rule

For a rational function limit at infinity where numerator and denominator degrees are equal, the limit is the ratio of the coefficients of the highest degree terms.

Comparison Test

A method of testing convergence by directly comparing the terms of a series to the terms of a known convergent or divergent series.

Root Test (Convergence Range)

According to the root test, absolute convergence is confirmed if the limit is between $0$ and $1$, but not including $1$.