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1 Algebraic and Analytic MethodsAreas

§1.2 Elementary Algebra

  1. §1.2(i) Binomial Coefficients
  2. §1.2(ii) Finite Series
  3. §1.2(iii) Partial Fractions
  4. §1.2(iv) Means

§1.2(i) Binomial Coefficients

In (1.2.1) and (1.2.3) k and n are nonnegative integers and kn. In (1.2.2), (1.2.4), and (1.2.5) n is a positive integer. See also §26.3(i).

1.2.1 (nk)=n!(nk)!k!=(nnk).

For complex z the binomial coefficient (zk) is defined via (1.2.6).

Binomial Theorem

1.2.2 (a+b)n=an+(n1)an1b+(n2)an2b2++(nn1)abn1+bn.
1.2.3 (n0)+(n1)++(nn)=2n.
1.2.4 (n0)(n1)++(1)n(nn)=0.
1.2.5 (n0)+(n2)+(n4)++(n)=2n1,

where is n or n1 according as n is even or odd.

In (1.2.6)–(1.2.9) k and m are nonnegative integers and z is complex.

1.2.6 (zk)=z(z1)(zk+1)k!=(1)k(z)kk!=(1)k(kz1k).
1.2.7 (z+1k)=(zk)+(zk1).
1.2.8 k=0m(z+kk)=(z+m+1m).
1.2.9 (z0)(z1)++(1)m(zm)=(1)m(z1m).

See also §26.3.

§1.2(ii) Finite Series

Arithmetic Progression

1.2.10 a+(a+d)+(a+2d)++(a+(n1)d)=na+12n(n1)d=12n(a+),

where = last term of the series = a+(n1)d.

Geometric Progression

1.2.11 a+ax+ax2++axn1=a(1xn)1x,

§1.2(iii) Partial Fractions

Let α1,α2,,αn be distinct constants, and f(x) be a polynomial of degree less than n. Then

1.2.12 f(x)(xα1)(xα2)(xαn)=A1xα1+A2xα2++Anxαn,


1.2.13 Aj=f(αj)kj(αjαk).


1.2.14 f(x)(xα1)n=B1xα1+B2(xα1)2++Bn(xα1)n,


1.2.15 Bj=f(nj)(α1)(nj)!,

and f(k) is the k-th derivative of f1.4(iii)).

If m1,m2,,mn are positive integers and degf<j=1nmj, then there exist polynomials fj(x), degfj<mj, such that

1.2.16 f(x)(xα1)m1(xα2)m2(xαn)mn=f1(x)(xα1)m1+f2(x)(xα2)m2++fn(x)(xαn)mn.

To find the polynomials fj(x), j=1,2,,n, multiply both sides by the denominator of the left-hand side and equate coefficients. See Chrystal (1959a, pp. 151–159).

§1.2(iv) Means

The arithmetic mean of n numbers a1,a2,,an is

1.2.17 A=a1+a2++ann.

The geometric mean G and harmonic mean H of n positive numbers a1,a2,,an are given by

1.2.18 G=(a1a2an)1/n,
1.2.19 1H=1n(1a1+1a2++1an).

If r is a nonzero real number, then the weighted mean M(r) of n nonnegative numbers a1,a2,,an, and n positive numbers p1,p2,,pn with

1.2.20 p1+p2++pn=1,

is defined by

1.2.21 M(r)=(p1a1r+p2a2r++pnanr)1/r,

with the exception

1.2.22 M(r)=0,
r<0 and a1a2an=0.
1.2.23 limrM(r) =max(a1,a2,,an),
1.2.24 limrM(r) =min(a1,a2,,an).

For pj=1/n, j=1,2,,n,

1.2.25 M(1) =A,
M(1) =H,


1.2.26 limr0M(r)=G.

The last two equations require aj>0 for all j.