Math 152 Midterm 2 Review: Sequences and Series

A set of vocabulary flashcards based on Professor Mavrea's Math 152 review notes for the second midterm, covering sequence and series convergence, power series, and Taylor polynomials.

Sequence Convergence

A sequence $\{a_n\}$ converges if its limit as $n \rightarrow \frac{}{}\text{infinity}$ is a finite number.

Divergence Test

A test used to determine if a series $\text{diverges}$; if $\text{limit}_{n \rightarrow \text{infinity}} a_n \neq 0$, then the series $\text{diverges}$. This test can never be used to show a series converges.

Ratio Test

A convergence test that calculates the limit of the ratio of successive terms, $|a_{n+1} / a_n|$, to determine if a series converges absolutely, diverges, or is inconclusive.

Limit Comparison Test

A method to determine the convergence or divergence of a series by comparing the limit of the ratio of its terms to the terms of a known series as $n \rightarrow \text{infinity}$.

Integral Test Hypotheses

The specific conditions that must be satisfied to use the integral test: the function $f(x)$ must be continuous, positive, and decreasing for $x \text{ in } [1, \text{infinity})$.

Partial Sum ($S_n$)

The sum of the first $n$ terms of an infinite series, defined as $S_n = \text{sum from } j=0 \text{ to } n \text{ of } a_j$.

Closed Formula

An explicit mathematical expression for the partial sum $S_n$ that does not involve summation notation.

Radius of Convergence

The distance $R$ from the center of a power series within which the series is guaranteed to converge.

Interval of Convergence

The set of all values of $x$ for which a power series converges, often found by applying the ratio test and checking behavior at the endpoints.

Harmonic Series

The series represented by the sum $\text{sum from } n=1 \text{ to } \text{infinity of } \frac{1}{n}$, which is known to diverge.

P-series

A series of the form $\text{sum from } n=1 \text{ to } \text{infinity of } \frac{1}{n^p}$, used frequently for comparison purposes in convergence tests.

Taylor Polynomial

An approximation of a function $f(x)$ using a polynomial of degree $n$ centered at a specific value such as $x = 1000$.

Maclaurin Series

A Taylor series expansion of a function centered specifically at $x = 0$.

Error Bound

The maximum possible difference between the actual value of a function and the value estimated by a Taylor polynomial.

Power Series

An infinite series of the form $\text{sum of } c_n(x-a)^n$, which functions as a polynomial with an infinite number of terms.