CHAPTER II LIMITS AND SERIES 4. Variables and sequences. In mathematics it is necessary to consider two very distinct kinds of numbers, or quantities. Numbers of the one kind do not change during the discussion of the problem in which they occur, and are called constant num- bers, or simply constants. The numbers 2, 5, ^, the first term of a progression, are constants. On the other hand, a number which takes different values in the discussion of a problem is called a variable. The number s, of the previous section is a variable. Moreover, the number n itself is a variable upon which the value of «„ depends. It is customary to use the first letters of the alphabet to denote constants, and the last letters to denote variables. In the problems in which variables occur the values which the variables may take are usually restricted to sets of numbers fol- lowing definite laws. Thus, in the infinite geometrical progres- sions the variable n may take the values 1, 2, 3, 4 . In the stick problem «„ took the values 1,1^,1|, 1^ . A set of numbers in which the successive numbers are deter- mined according to some definite law is called a sequence. A variable which takes the successive values of a sequence is said to run through the sequence. The variable «„ in the stick problem runs through the sequence 1, 1^, 1|, 1^ . Many operations of arithmetic lead us to sequences. For example, the attempt to reduce the common fraction ^ to a decimal leads us to the sequence .0, .00, .000, "* *, whose terms form the successive approximations to the value of the fraction. Again, the process of root extraction will lead to the sequence ^ ^ ^^ ^^ 1.4142, . ., 10 LIMITS AND SERIES 11 whose terms form the successive approximations to the value of V2. The extraction of any real root of any given number will lead to a definite sequence. Another very important example of a sequence is the set of numbers 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, -, which represent the successive stages in the attempt to find an approximate value of the ratio of the circumference of a circle to its diameter. In all the examples of sequences which have just been given there exists a fixed number such that the difference between it and the numbers of the sequence becomes smaller and smaller. When a variable x runs through the numbers of such a sequence, the fixed number a is called the limit of the variable, in accordance with the following definition: A fixed number a is called the limit of a variable x if, as x runs through the numbers of a sequence, the difference, a-x becomes and remains numerically smaller than any number that can be assigned. The relation between a variable x and its limit a is written lim x = a and is read " the limit of x equals a." The variable x may approach its limit in various ways. The values taken by x may be always less than a, or always greater than a, or sometimes less and sometimes greater. In the stick problem the values taken by »„ were always less than the limit 2; a variable which runs through the sequence U, 2y U, 2^, is always greater than its limit 2, while, if we consider the sum of n terms of the series 1-.1+.01-.001+.0001----, whose terms are the successive powers of .1 with signs alter- nately plus and minus, we obtain the sequence 1, .9, .91, .909, .9091, .90909, 12 MATHEMATICAL THEORY OF INVESTMENT The first, third, fifth, and indeed all the odd-numbered terms of this sequence are greater than ^, while the even-numbered terms are less than the same fraction, which is the limit of the variable thai runs through the sequence. Whatever may be the relation between a and a; as a: is approaching a, we write a—x to denote the numerical, as opposed to the algebraic, value of the difference between a and x. According to this notation [ 3 — 51 and 15 — 31 have the same value 2. A quantity inclosed between such bars is always positive. In order that x shall approach the limit a, it is necessary and sufficient that | a — x \ should become and remain less than e, however small e may be chosen. The following theorems, for which demonstrations may be found in books on algebra and analysis, are of great use in many branches of mathematics: I. If a variable x always increases but never takes values greater than a fixed number A, it approaches a limit which is either A or some number less than A. Similarly, if x always decreases but never takes values smaller than a fixed number A', it approaches a limit which is either A' or some number greater than A'. II.If two variables are always equal and each approaches a limit, their limits are equal; i:e. if x = y, Urn x= a, and limy = b, then a = b. III. The limit of the product of a constant and a variable is equal to the product of the constant and the limit of the variable. In symbols, lim ex = c lim x. IV. The limit of the sum, difference, or product of two variables is equal to the sum, difference, or product of their limits. Or, lim (x + y) = lim x + lim y, lim (x — y) = lim x — lim y, lim(xy) == lim x lim y. V. The limit of the quotient of two variables is equal to the quo- tient of their limits, provided the limit of the divisor is different . . ,. x limx from zero, i.e.. lim-=——— y hmy LIMITS AND SERIES 13 VIThe limit of the reciprocal of a variable whose limit is different from zero is equal to the reciprocal of the limit of the variable. Insymbols, lim -^ = ~^~^ VII.The limit of a power of a variable is equal to the power of the limit, provided we have not at the same time limx=0 and n < 0 ; or, lim of = (lim x)°. In solving problems for the determination of limits it is usually necessary to perform some algebraic reduction upon the expression whose limit is to be found, and then apply one or more of the foregoing theorems. For example.^uppose it is required to find the limit of the expression ^^ as x becomes indefinitely large. If both numerator and denominator be divided through by a-, we have the identity . s+o 3a-+5___s_ 4^+2~4^2' Applying in succession Theorems II, V, IV,and VI,we find 3+5 innlt^lim——— 4a;+2 4_^ x limf3+5) \ x/