June 30, 2026 - Calculus 2 - Series and Convergence Lecture Notes

A set of 20 vocabulary-style flashcards covering series convergence and divergence tests based on the lecture transcript.

Geometric Series

A type of series that can be solved using a specific formula when the form is $a \cdot r^{n-1}$ and the index starts at $1$.

Common Ratio ($r$)

The value in a geometric series that must be between $-1$ and $1$ ($|r| < 1$) for the series to converge.

Telescoping series

A series in which terms in the middle cancel out, leaving only a few terms at the beginning and the end.

Partial sum ($s_k$)

The sum of a specified number of terms in a series, which determines convergence based on its limit as $k \rightarrow \infty$.

Divergence test

A test stating that if the limit of the sequence $a_n$ is anything other than zero, or is plus/minus infinity, then the series diverges.

Inconclusive

The result of the divergence test when the limit of the sequence is zero; it does not confirm if the series converges or diverges.

Harmonic series

The series represented by the sum of $\frac{1}{n}$, which is divergent even though the limit of its terms is zero.

Integral test

A method to determine convergence or divergence by comparing a series to the improper integral of a related function $f(x)$. child

Continuous

One of the three requirements for the integral test, specifying that the function $f$ must not have breaks in the interval of integration.

Decreasing

A mandatory condition for the integral test and the divergence of terms, meaning the values of the terms get smaller as $n$ increases.

Non-negative

A criteria for the integral test stating that the terms $a_n$ of the series must be greater than or equal to zero.

P-series

A series of the form $\sum \frac{1}{n^p}$, where the convergence depends solely on the value of the exponent $P$.

Convergence of P-series

The condition where a P-series converges if the exponent $P$ is strictly greater than $1$ ($P > 1$).

Divergence of P-series

The condition where a P-series diverges if the exponent $P$ is less than or equal to $1$ ($P \le 1$).

Comparison test

A test that compares a given series to another known series to determine convergence or divergence based on their relative sizes.

Monotone (Monotonic)

A property of a sequence that is either eventually increasing or eventually decreasing.

Bounded

A property of a sequence where there exists a number that the values of the partial sums never exceed.

Li talls

The student's phonetic spelling or the instructor's shorthand for l'Hôpital's Rule, used to evaluate limits like those of rational functions.

Index manipulation

The process of changing the starting point ($n$) or the exponent of a series term to fit a standard formula like the geometric series formula.

Limit of the partial sums

The value to which a series converges if the sequence of its partial sums approaches a finite number.