July 01, 2026 - Calculus 2 - Comparison and Alternating Series Tests

Vocabulary and concepts regarding infinite series convergence tests including p series, comparison tests, and alternating series.

P series

A series of the form $\sum_{n=1}^{\infty} \frac{1}{n^p}$ which converges if $p > 1$ and diverges otherwise.

Comparison Test (Convergence Condition)

If the series of $b_n$ converges and the inequality $a_n \le b_n$ holds for some capital $N$, then the series of $a_n$ also converges.

Comparison Test (Divergence Condition)

If the series of $b_n$ diverges and the inequality $a_n \ge b_n$ holds after a certain point, then the series of $a_n$ also diverges.

Capital N

A value used to indicate that sequences can behave differently at the beginning, but must follow a specific inequality after a certain point.

Harmonic series

A specific name from physics for the p series where $p = 1$, which is known to diverge.

n (notation)

The variable used when talking about integers in convergence and divergence problems.

C (notation)

A variable that represents any real number, as opposed to an integer.

Geometric series

A series involving terms like $\frac{1}{2^n}$ that converges if the common ratio is between $1$ and $-1$.

Divergence test

A test that determines a series diverges if the limit of the sequence terms does not approach zero, though it is inconclusive if the limit is zero.

Limit comparison test

A technique used when a simple inequality cannot be set up, involving taking the limit of the ratio of two sequences as $n$ goes to infinity.

LCT Case 1

If $\lim_{n \to \infty} \frac{a_n}{b_n} = C$, where $C$ is a constant and $C \neq 0$, then both sequences either converge or diverge.

LCT Case 2

If $\lim_{n \to \infty} \frac{a_n}{b_n} = 0$ and the denominator series converges, then the numerator series also converges.

LCT Case 3

If $\lim_{n \to \infty} \frac{a_n}{b_n} = \infty$ and the denominator series diverges, then the numerator series also diverges.

Alternating series

A series that moves back and forth between positive and negative values, typically denoted by $(-1)^n$ or $(-1)^{n+1}$.

Telescopic series

A series where terms in the middle cancel out, causing the series to collapse like a telescope.

Alternating series test (Condition 1)

Requires the sequence of terms (without the negative sign) to be decreasing, meaning $a_{n+1} \le a_n$.

Alternating series test (Condition 2)

Requires the limit of the sequence terms $a_n$ to approach zero as $n$ goes to infinity.

Taylor and McLaren

Specific types of series that form a whole module following the study of general convergence tests.

Power series

The topic to be covered in the last two weeks of the module based on the lecture schedule.

Integral test

A test used to prove the convergence behavior of p series.