436. Infinite Geometrical Series. When r < 1, every term of a geometrical series is numerically smaller than the preceding term, as in 1, 4, 4, 16. It is 1 8, evident that the terms decrease in value as the number of terms increases, and that as more and more terms are included, the nth term, ar-1, becomes more and more nearly equal to zero. Hence, by taking a sufficient number of terms, we can make the nth term as small as we please, although it can never become exactly zero. As we have seen, by taking a sufficient number of terms, we can make a", and consequently the fraction as small as we please; the greater the number of terms, the nearer the value of this fraction approaches 0, and consequently the sum of the terms approaches a 1 And since we can take as many terms as we please, be made to become less in absolute value than any positive number that may be assigned; that is, the difference between the fraction and zero will become less than any value that can be expressed by any finite number, however small. We there the sum of an unlimited (i.e., infinite) number of terms. The derivation of this formula should be reviewed in connection with the next chapter, Art. 454. 1. Find the sum of the series 1,1,1, Here The sum really is 2 a 1 = 2. ; and since this neglected fraction, however small, is somewhat greater than absolute 0, the required sum is just that much less than 2. Since this difference can be made less than any number that may be assigned, however small, we are not able to give it any other value than zero, and it is therefore ignored. 437. 1. What kind of series is 2, 4, 6, 8, reciprocal of each term, we have 1, 1, 1, 1, metical series? Has it a common difference? 3 5 ... ? Taking the Is this an arith 2. Is 4,,, t ... an arithmetical progression? Taking the reciprocal of each term, we have 1, 4, 4, 4, ... What kind of series is this? 438. A series of numbers, the reciprocals of which are in arithmetical progression, is called a Harmonic Progression, or a Harmonic Series. Thus, 1, 2, 3, 4, ... is a harmonic progression, since the reciprocals, are in arithmetical progression. If a violin string be made to vibrate in parts, i.e., halves, thirds, quarters, etc., notes called harmonics are produced. The relative lengths of vibrating parts are expressed numerically by the reciprocals of the natural numbers. 1. Write the harmonic series corresponding to the arithmetical progressions 3, 5, 7, ..., and a, a +d, a + 2 d, ........ .... 2. Find the 10th term of the harmonic progression 8, 4, 3. Find the 9th term of the series 1, }, }, 4. Find the 10th term of the series .... 1 1 1 439. If H is the harmonic mean between a and b, then, by 1 1 1 definition, a H' b are in arithmetical progression. Hence 1. Insert a harmonic mean between 3 and 7. and 2. Insert 3 harmonic means between 1 = a + (n − 1)d, or 10 = 2 + 4 d. .. d = 2. Hence, the arithmetical series is 2, 4, 6, 8, 10, and the three required means are,, and . 3. Find the harmonic mean between 4 and - 3. Between and 1. 4. Insert 2 harmonic means between 5 and 2. Between m and n. 1 1 1' .... The nth term. 5. Find the 7th term of 1, 4, 8, 8, 6. Insert 4 harmonic means between .5 and -7. 7. Write the first 6 terms of a harmonic progression whose 3d and 4th terms are 2 and 14. 8. Find the harmonic mean between m n and m + n. 9. What are the two numbers whose harmonic mean is 3 and whose arithmetical mean is 4? 10. If m, n, r, are in harmonic progression, prove that r(m − n) = m(n − r). 11. If the harmonic mean between m and n is r, prove that 12. The first two terms of a harmonic series are 5 and 6. What term will be 30? 13. The arithmetical mean between two numbers is 3, and the harmonic mean is . What are the numbers? 14. Show that the geometrical mean of any two numbers is also the geometrical mean between their arithmetical and harmonic means. 15. What number must be added to each of three given numbers a, b, c, that the three results may be in harmonic progression? 16. If a, b, and c are in harmonic progression, prove that LIMITS, INFINITY, INDETERMINATE FRACTIONS 440. As already stated (Art. 409), a number that may have different values (usually an indefinite number of them) in the same expression or equation is called a Variable; on the other hand, a number whose value is fixed is called a Constant. It is customary to represent variables by the last letters of the alphabet, and constants by the first letters. 441. It has also been stated (Art. 410) that one variable is said to be a Function of another when they are so related that the value of the first depends upon that of the second. Thus, a variable y is a function of a variable x, given to x there corresponds a definite value of y. is x a function of y? if to every value that is Under what conditions In like manner, the algebraic expression x2 + x + 1 is said to be a function of x, since its value depends upon that of x. If the value of an expression depends upon the values of x and y, the expression is said to be a function of these variables. 442. For brevity, the symbol f(x), read "function of x," is commonly used to denote a function of x. Thus, any algebraic expression containing x, as x2 − 3 x + 2, may be conveniently denoted by f(x). If two different functions of ≈ occur in the same operation, they may be distinguished as ƒ(x) and F(x). 443. To denote what a function becomes when particular values are substituted for the variable involved, we substitute the same value for the variable in the functional symbol. Thus, if f(x) = x2 - 2x + 1, then f(a) means that function when x is replaced by a, that is, f(a) = a2 — 2a+1. Again, if ƒ (x) = x2 − 5 x + 6, then f(2) 4 - 10 + 6 = 0. |