1

–

6

1

–

2

268 inference

√n2+ 100n

1

–

4

1

–

3

1

–

2

(This can be proved by truncating the M

ACLAURIN

SERIES

:

(1 + x)n= 1 + nx + n(n– 1)x2

+ n(n– 1)(n– 2)x3+ …,

One can also show that the inequality is also true for

–1 < x< 0).

Cauchy-Schwarz inequality: For any two

VECTORS

a

and b, their

DOT PRODUCT

satisfies:

|a· b| ≤|a| ×|b|

Thus, for any two lists of numbers a1,a2,…,anand

b1,b2,…,bnwe have:

Weierstrass’s product inequality: For any list of num-

bers a1,a2,…,anwith 0 ≤ak≤1 for all k, we have:

(1 – a1)(1 – a2)…(1 – an) ≥1 + a1+ a2+…+an

(This can be proved by

INDUCTION

on the number of

elements in the list.)

Napier’s inequality: For any two positive real numbers

aand bwe have:

Exponential inequalities: For a positive real number x

and a real number c, we have:

xc< 1 + c(x– 1) if 0 < c< 1

xc> 1 + c(x– 1) if c> 1

(This is a generalization of Bernoulli’s inequality.)

Isoperimetric inequality: For any closed geometric fig-

ure in the plane with perimeter P, its area Ais less than

the area of a circle of the same perimeter:

Equality holds if, and only if, that figure is a circle.

A mathematical statement that two quantities or

expressions are never equal is called an inequation. For

instance, the statement 3 ≠5 is an inequation.

Mathematicians and physicists often write a>> bif

ais significantly larger than band a<< bis ais signifi-

cantly smaller. For example, ≈nif n>> 0.

There is a joke among mathematicians to use the sym-

bol ≤≥ to mean “less than, greater than, or possibly

equal to” when one is not sure of the numerical answer

to a problem.

See also

ARITHMETIC

-

GEOMETRIC MEAN INEQUAL

-

ITY

;

ISOPERIMETRIC PROBLEM

;

TRIANGLE INEQUALITY

.

inference In logic, the general process of developing

an

ARGUMENT

to draw a conclusion from a set of

premises is called inference. The process could be

deductive or inductive.

In statistics, inference is the process of coming to a

conclusion about a population based on a study of a

sample. Sometimes the conclusion itself is called an

inference. Inferential statistics is the science of making

inferences and predictions about a population based on

numerical information gathered from a sample.

See also

DEDUCTIVE

/

INDUCTIVE REASONING

;

POPU

-

LATION AND SAMPLE

;

STATISTICS

:

INFERENTIAL

.

infinite product The product of an infinite number

of factors, a1×a2×a3×…, is called an infinite product.

The nth number in the product is called the nth term of

the product, and the product of the first nterms, Pn=

a1×a2×…×anis called the nth partial product. In

1812 C

ARL

F

RIEDRICH

G

AUSS

introduced the notation

for an infinite product (and for the nth

partial product).

An infinite product might have a value of zero (1 ×

×××…, for example), might be infinite in

value (1 ×2 ×3 ×4 ×…, for example), or could oscillate

in value (1 ×(–1) ×1 ×(–1) ×…, for instance). Only if

the partial products Pnapproach a finite nonzero value

Las n→∞is the infinite product said to converge (to

the value L). Otherwise the infinite product diverges.

For example, the infinite product has nth

partial product: