| Term | Definition |
|---|---|
| converges | A series converges when the sequence of partial sums approaches a finite limit as n approaches infinity. |
| diverges | A series diverges when the sequence of partial sums does not approach a finite limit as the number of terms increases indefinitely. |
| limit | The value that a function approaches as the input approaches some value, which may or may not equal the function's value at that point. |
| nth partial sum | The sum of the first n terms of a series. |
| sequence of partial sums | The sequence formed by successive partial sums of a series, where each term is the sum of the first n terms. |
| series | A sum of the terms of a sequence, often written as the sum of infinitely many terms. |
| Term | Definition |
|---|---|
| alternating series | A series whose terms alternate in sign, typically written in the form Σ(-1)^n * a_n where a_n > 0. |
| alternating series error bound | A method for estimating the maximum error between a partial sum and the actual sum of a convergent alternating series, equal to the absolute value of the first omitted term. |
| alternating series test | A convergence test that determines whether an alternating series converges based on whether its terms decrease in absolute value and approach zero. |
| converges | A series converges when the sequence of partial sums approaches a finite limit as n approaches infinity. |
| partial sum | The sum of the first n terms of a series, denoted S_n. |
| series | A sum of the terms of a sequence, often written as the sum of infinitely many terms. |
| Term | Definition |
|---|---|
| centered at | The point around which a Taylor polynomial is constructed; the point where the polynomial and function share the same value and derivatives. |
| coefficient | The numerical factor in front of a term in a polynomial, calculated as the nth derivative of the function divided by n factorial. |
| degree | The highest power of the variable in a polynomial term. |
| derivative | The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point. |
| factorial | The product of all positive integers up to a given number, denoted as n!, used in the denominator of Taylor polynomial coefficients. |
| function approximation | Using a simpler function, such as a polynomial, to estimate the values of a more complex function. |
| interval | A connected set of real numbers, typically expressed as a range between two endpoints. |
| Taylor polynomial | A finite polynomial that approximates a function, formed by taking a partial sum of the Taylor series for that function. |
| Term | Definition |
|---|---|
| alternating series error bound | A method for estimating the maximum error between a partial sum and the actual sum of a convergent alternating series, equal to the absolute value of the first omitted term. |
| error bound | A maximum value that represents how far a Taylor polynomial approximation can deviate from the actual function value. |
| Lagrange error bound | A formula that provides the maximum possible error when using a Taylor polynomial to approximate a function value. |
| Taylor polynomial approximation | A polynomial function used to approximate the value of a function near a specific point. |
| Term | Definition |
|---|---|
| interval of convergence | The set of all x-values for which a power series converges, determined by testing the radius of convergence and checking the endpoints. |
| power series | An infinite series of the form Σ(aₙ(x-c)ⁿ) where aₙ are coefficients, x is a variable, and c is the center of the series. |
| radius of convergence | The value that determines the distance from the center of a power series within which the series converges. |
| ratio test | A convergence test used to determine the radius of convergence of a power series by examining the limit of the ratio of consecutive terms. |
| Taylor series | A power series representation of a function that converges to that function over an open interval with positive radius of convergence. |
| term-by-term differentiation | The process of differentiating a power series by differentiating each term individually, which preserves the radius of convergence. |
| term-by-term integration | The process of integrating a power series by integrating each term individually, which preserves the radius of convergence. |
| Term | Definition |
|---|---|
| geometric series | A series where each term is a constant multiple of the previous term, expressed in the form ∑_{n=0}^{∞} a r^{n}. |
| Maclaurin series | A special case of a Taylor series where the function is expanded around the point x = 0. |
| Taylor polynomial | A finite polynomial that approximates a function, formed by taking a partial sum of the Taylor series for that function. |
| Taylor series | A power series representation of a function that converges to that function over an open interval with positive radius of convergence. |
| Term | Definition |
|---|---|
| geometric series | A series where each term is a constant multiple of the previous term, expressed in the form ∑_{n=0}^{∞} a r^{n}. |
| power series | An infinite series of the form Σ(aₙ(x-c)ⁿ) where aₙ are coefficients, x is a variable, and c is the center of the series. |
| term-by-term differentiation | The process of differentiating a power series by differentiating each term individually, which preserves the radius of convergence. |
| term-by-term integration | The process of integrating a power series by integrating each term individually, which preserves the radius of convergence. |
| Term | Definition |
|---|---|
| constant ratio | The fixed multiplicative factor between successive terms in a geometric series, denoted as r. |
| converges | A series converges when the sequence of partial sums approaches a finite limit as n approaches infinity. |
| diverges | A series diverges when the sequence of partial sums does not approach a finite limit as the number of terms increases indefinitely. |
| geometric series | A series where each term is a constant multiple of the previous term, expressed in the form ∑_{n=0}^{∞} a r^{n}. |
| Term | Definition |
|---|---|
| converges | A series converges when the sequence of partial sums approaches a finite limit as n approaches infinity. |
| diverges | A series diverges when the sequence of partial sums does not approach a finite limit as the number of terms increases indefinitely. |
| nth term test | A test for divergence that examines whether the limit of the nth term of a series equals zero; if the limit is not zero, the series diverges. |
| series | A sum of the terms of a sequence, often written as the sum of infinitely many terms. |
| Term | Definition |
|---|---|
| converges | A series converges when the sequence of partial sums approaches a finite limit as n approaches infinity. |
| diverges | A series diverges when the sequence of partial sums does not approach a finite limit as the number of terms increases indefinitely. |
| integral test | A method for determining whether an infinite series converges or diverges by comparing it to an improper integral. |
| series | A sum of the terms of a sequence, often written as the sum of infinitely many terms. |
| Term | Definition |
|---|---|
| alternating harmonic series | The infinite series 1 - 1/2 + 1/3 - 1/4 + ..., which converges to ln(2). |
| converges | A series converges when the sequence of partial sums approaches a finite limit as n approaches infinity. |
| diverges | A series diverges when the sequence of partial sums does not approach a finite limit as the number of terms increases indefinitely. |
| geometric series | A series where each term is a constant multiple of the previous term, expressed in the form ∑_{n=0}^{∞} a r^{n}. |
| harmonic series | The infinite series 1 + 1/2 + 1/3 + 1/4 + ..., which diverges despite having terms that approach zero. |
| p-series | An infinite series of the form 1 + 1/2^p + 1/3^p + 1/4^p + ..., which converges when p > 1 and diverges when p ≤ 1. |
| series | A sum of the terms of a sequence, often written as the sum of infinitely many terms. |
| Term | Definition |
|---|---|
| comparison test | A method for determining convergence or divergence of a series by comparing it to another series whose convergence is known. |
| converges | A series converges when the sequence of partial sums approaches a finite limit as n approaches infinity. |
| diverges | A series diverges when the sequence of partial sums does not approach a finite limit as the number of terms increases indefinitely. |
| limit comparison test | A method for determining convergence or divergence of a series by comparing the limit of the ratio of its terms to those of another series. |
| series | A sum of the terms of a sequence, often written as the sum of infinitely many terms. |
| Term | Definition |
|---|---|
| alternating series | A series whose terms alternate in sign, typically written in the form Σ(-1)^n * a_n where a_n > 0. |
| alternating series test | A convergence test that determines whether an alternating series converges based on whether its terms decrease in absolute value and approach zero. |
| converges | A series converges when the sequence of partial sums approaches a finite limit as n approaches infinity. |
| diverges | A series diverges when the sequence of partial sums does not approach a finite limit as the number of terms increases indefinitely. |
| Term | Definition |
|---|---|
| converges | A series converges when the sequence of partial sums approaches a finite limit as n approaches infinity. |
| diverges | A series diverges when the sequence of partial sums does not approach a finite limit as the number of terms increases indefinitely. |
| ratio test | A convergence test used to determine the radius of convergence of a power series by examining the limit of the ratio of consecutive terms. |
| series | A sum of the terms of a sequence, often written as the sum of infinitely many terms. |
| Term | Definition |
|---|---|
| absolutely convergent | A series that converges when all terms are replaced by their absolute values. |
| conditionally convergent | A series that converges but does not converge absolutely; the series converges only because of the signs of its terms. |
| converges | A series converges when the sequence of partial sums approaches a finite limit as n approaches infinity. |
| diverges | A series diverges when the sequence of partial sums does not approach a finite limit as the number of terms increases indefinitely. |
| series | A sum of the terms of a sequence, often written as the sum of infinitely many terms. |