Calculus Volume 2

Key Terms

binomial series
the Maclaurin series for $f(x)=(1+x)r;f(x)=(1+x)r;$ it is given by
$(1+x)r=∑n=0∞(rn)xn=1+rx+r(r−1)2!x2+⋯+r(r−1)⋯(r−n+1)n!xn+⋯(1+x)r=∑n=0∞(rn)xn=1+rx+r(r−1)2!x2+⋯+r(r−1)⋯(r−n+1)n!xn+⋯$ for $|x|<1|x|<1$
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 $ff$ at 0 is the nth Maclaurin polynomial for $ff$
Maclaurin series
a Taylor series for a function $ff$ at $x=0x=0$ is known as a Maclaurin series for $ff$
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 $∑n=0∞cnxn∑n=0∞cnxn$ is a power series centered at $x=0;x=0;$ a series of the form $∑n=0∞cn(x−a)n∑n=0∞cn(x−a)n$ is a power series centered at $x=ax=a$
if there exists a real number $R>0R>0$ such that a power series centered at $x=ax=a$ converges for $|x−a| and diverges for $|x−a|>R,|x−a|>R,$ then R is the radius of convergence; if the power series only converges at $x=a,x=a,$ the radius of convergence is $R=0;R=0;$ if the power series converges for all real numbers x, the radius of convergence is $R=∞R=∞$
Taylor polynomials
the nth Taylor polynomial for $ff$ at $x=ax=a$ is $pn(x)=f(a)+f′(a)(x−a)+f″(a)2!(x−a)2+⋯+f(n)(a)n!(x−a)npn(x)=f(a)+f′(a)(x−a)+f″(a)2!(x−a)2+⋯+f(n)(a)n!(x−a)n$
Taylor series
a power series at a that converges to a function $ff$ on some open interval containing a
Taylor’s theorem with remainder
for a function $ff$ and the nth Taylor polynomial for $ff$ at $x=a,x=a,$ the remainder $Rn(x)=f(x)−pn(x)Rn(x)=f(x)−pn(x)$ satisfies $Rn(x)=f(n+1)(c)(n+1)!(x−a)n+1Rn(x)=f(n+1)(c)(n+1)!(x−a)n+1$
for some c between x and a; if there exists an interval I containing a and a real number M such that $|f(n+1)(x)|≤M|f(n+1)(x)|≤M$ for all x in I, then $|Rn(x)|≤M(n+1)!|x−a|n+1|Rn(x)|≤M(n+1)!|x−a|n+1$
term-by-term differentiation of a power series
a technique for evaluating the derivative of a power series $∑n=0∞cn(x−a)n∑n=0∞cn(x−a)n$ by evaluating the derivative of each term separately to create the new power series $∑n=1∞ncn(x−a)n−1∑n=1∞ncn(x−a)n−1$
term-by-term integration of a power series
a technique for integrating a power series $∑n=0∞cn(x−a)n∑n=0∞cn(x−a)n$ by integrating each term separately to create the new power series $C+∑n=0∞cn(x−a)n+1n+1C+∑n=0∞cn(x−a)n+1n+1$