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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(r1)2!x2++r(r1)(rn+1)n!xn+(1+x)r=n=0(rn)xn=1+rx+r(r1)2!x2++r(r1)(rn+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=0cnxnn=0cnxn is a power series centered at x=0;x=0; a series of the form n=0cn(xa)nn=0cn(xa)n is a power series centered at x=ax=a
radius of convergence
if there exists a real number R>0R>0 such that a power series centered at x=ax=a converges for |xa|<R|xa|<R and diverges for |xa|>R,|xa|>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)(xa)+f(a)2!(xa)2++f(n)(a)n!(xa)npn(x)=f(a)+f(a)(xa)+f(a)2!(xa)2++f(n)(a)n!(xa)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)!(xa)n+1Rn(x)=f(n+1)(c)(n+1)!(xa)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)!|xa|n+1|Rn(x)|M(n+1)!|xa|n+1
term-by-term differentiation of a power series
a technique for evaluating the derivative of a power series n=0cn(xa)nn=0cn(xa)n by evaluating the derivative of each term separately to create the new power series n=1ncn(xa)n1n=1ncn(xa)n1
term-by-term integration of a power series
a technique for integrating a power series n=0cn(xa)nn=0cn(xa)n by integrating each term separately to create the new power series C+n=0cn(xa)n+1n+1C+n=0cn(xa)n+1n+1
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