Key Terms - Calculus Volume 2

Key Terms

binomial series: the Maclaurin series for $f (x) = {(1 + x)}^{r};$ it is given by
${(1 + x)}^{r} = \sum_{n = 0}^{\infty} (\begin{matrix} r \\ n \end{matrix}) x^{n} = 1 + r x + \frac{r (r - 1)}{2!} x^{2} + ⋯ + \frac{r (r - 1) ⋯ (r - n + 1)}{n!} x^{n} + ⋯$ for $| x | < 1$

interval of convergence: the set of real numbers x for which a power series converges

Maclaurin polynomial: a Taylor polynomial centered at 0; the nth Taylor polynomial for $f$ at 0 is the nth Maclaurin polynomial for $f$

Maclaurin series: a Taylor series for a function $f$ at $x = 0$ is known as a Maclaurin series for $f$

nonelementary integral: an integral for which the antiderivative of the integrand cannot be expressed as an elementary function

power series: a series of the form $\sum_{n = 0}^{\infty} c_{n} x^{n}$ is a power series centered at $x = 0;$ a series of the form $\sum_{n = 0}^{\infty} c_{n} {(x - a)}^{n}$ is a power series centered at $x = a$

radius of convergence: if there exists a real number $R > 0$ such that a power series centered at $x = a$ converges for $| x - a | < R$ and diverges for $| x - a | > R,$ then R is the radius of convergence; if the power series only converges at $x = a,$ the radius of convergence is $R = 0;$ if the power series converges for all real numbers x, the radius of convergence is $R = \infty$

Taylor polynomials: the nth Taylor polynomial for $f$ at $x = a$ is $p_{n} (x) = f (a) + f^{'} (a) (x - a) + \frac{f^{″} (a)}{2!} {(x - a)}^{2} + ⋯ + \frac{f^{(n)} (a)}{n!} {(x - a)}^{n}$

Taylor series: a power series at a that converges to a function $f$ on some open interval containing a

Taylor’s theorem with remainder: for a function $f$ and the nth Taylor polynomial for $f$ at $x = a,$ the remainder $R_{n} (x) = f (x) - p_{n} (x)$ satisfies $R_{n} (x) = \frac{f^{(n + 1)} (c)}{(n + 1)!} {(x - a)}^{n + 1}$
for some c between x and a; if there exists an interval I containing a and a real number M such that $| f^{(n + 1)} (x) | \leq M$ for all x in I, then $| R_{n} (x) | \leq \frac{M}{(n + 1)!} {| x - a |}^{n + 1}$

term-by-term differentiation of a power series: a technique for evaluating the derivative of a power series $\sum_{n = 0}^{\infty} c_{n} {(x - a)}^{n}$ by evaluating the derivative of each term separately to create the new power series $\sum_{n = 1}^{\infty} n c_{n} {(x - a)}^{n - 1}$

term-by-term integration of a power series: a technique for integrating a power series $\sum_{n = 0}^{\infty} c_{n} {(x - a)}^{n}$ by integrating each term separately to create the new power series $C + \sum_{n = 0}^{\infty} c_{n} \frac{{(x - a)}^{n + 1}}{n + 1}$