Calculus II Quiz 3 Review (9.1-9.6)

Flashcards covering sequences, series, and convergence tests (Section 9.1-9.6) from the MTH 126 lecture transcript.

Sequence Convergence

A sequence $\{a_n\}$ converges if the limit as $n \rightarrow \text{∞}$ exists and is a finite number $L$. If the limit does not exist or is infinite, the sequence diverges.

Arithmetic Sequence nth Term

The formula for a sequence where each term increases by a constant difference, such as $1, 6, 11, 16...$, where $d=5$ and the general term is $a_n = 5n - 4$.

Geometric Sequence nth Term

The formula for a sequence where each term is multiplied by a constant ratio, such as $3, 9, 27, 81...$, resulting in the expression $a_n = 3^n$.

Geometric Series Test

A series of the form $\sum_{n=0}^{\text{∞}} ar^n$ converges if $|r| < 1$ and diverges if $|r| \text{≥} 1$. If it converges, the sum is $S = \frac{a}{1 - r}$.

Telescoping Series

A series where the partial sums reduce to a fixed number of terms because intermediate terms cancel out, such as $\sum_{n=1}^{\text{∞}} \frac{12}{n(n+2)}$.

Integral Test

A convergence test requiring a function $f(x)$ to be positive, continuous, and decreasing for $x \text{≥} 1$. The series $\sum a_n$ converges if and only if the improper integral $\int_1^{\text{∞}} f(x)dx$ converges.

Direct Comparison Test

A test where terms of a series $a_n$ are compared to a known series $b_n$. If $0 < a_n \text{≤} b_n$ and $\sum b_n$ converges, then $\sum a_n$ converges. If $0 < b_n \text{≤} a_n$ and $\sum b_n$ diverges, then $\sum a_n$ diverges.

Limit Comparison Test

A test used to determine convergence by calculating $L = \text{lim}_{n \rightarrow \text{∞}} \frac{a_n}{b_n}$. If $L$ is a finite positive value ($L > 0$), then both $\sum a_n$ and $\sum b_n$ either converge or diverge together.

p-Series Test

A series of the form $\sum \frac{1}{n^p}$ converges if $p > 1$ and diverges if $p \text{≤} 1$.

Alternating Series Test

A series $\sum (-1)^n a_n$ converges if the limit of the terms is zero ($\text{lim}_{n \rightarrow \text{∞}} a_n = 0$) and the sequence of terms is non-increasing ($a_{n+1} \text{≤} a_n$).

Absolute Convergence

A series $\sum a_n$ is said to converge absolutely if the series of absolute values $\sum |a_n|$ converges.

Conditional Convergence

A series $\sum a_n$ is conditionally convergent if the series $\sum a_n$ converges, but the series of its absolute values $\sum |a_n|$ diverges.

nth-Term Test for Divergence

A test stating that if \text{lim}_{n \rightarrow \text{∞}} a_n \text{≠} 0, then the series $\sum a_n$ must diverge.

Ratio Test

A test involving the limit $L = \text{lim}_{n \rightarrow \text{∞}} |\frac{a_{n+1}}{a_n}|$. The series converges absolutely if $L < 1$, diverges if $L > 1$, and is inconclusive if $L = 1$.

Root Test

A test involving the limit L = \text{lim}_{n \rightarrow \text{∞}} \text{^}n\text{√}{|a_n|}. The series converges absolutely if $L < 1$, diverges if $L > 1$, and is inconclusive if $L = 1$.