The above will be the coefficients for any binomial square root (Art. 133.); hence the square root of 2 is actually expressed in the preceding series, if we make x=1. 1 Then (1+1)*=1+1— &c. The square root of 3 is " 2.4 expressed by the same series, when we make x=2, &c. 2. Required the cube root of (a+b) or its equal, a(1+4). quantity that we can put proper values to a and b. This expresses the cube root of any binomial quantity, or any into a binomial form, by giving the For example, required the cube root Here a=8, b=2. Then a*=2, of 10, or its equal, 8+2. 1 3. Expand into a series, or its equal, (a—b)—1. a 4. Expand (@a+¿2)ainto a series, or its equal, a(1+23). (Art. 136.) An infinite scries is a continued rank, or progression of quantities in regular order, in respect both to magnitudes and signs, and they usually arise from the division of one quantity by another. The roots of imperfect powers, as shown by the examples in the last article, produce one class of infinite series. Some of the examples under (Art. 121.) show the geometrical infinite series. Examples in common division may produce infinite series for quotients; or, in other words, we may say the division is continuous. Thus, 10 divided by 3, and carried out in decimals, gives 3.3333, &c., without end, and the sum of such a series is 3. (Art. 121.) (Art. 137.) Two series may appear very different, which arise from the same source; thus 1, divided by 1+a, gives, as we may see, by actual division, as follows: 1+a)1 1+a (1—a-+-a2—a3-+-a, &c. without end. a2 ر These two quotients appear very different, and in respect to single terms are so; but in these divisions there is always a remainder, and either quotient is incomplete without the remainder for a numerator and the divisor for a denominator, and when these are taken into consideration the two quotients will be equal. We may clearly illustrate this by the following example :Divide 3 by 1+2, the quotient is manifestly 1; but suppose them literal quantities, and the division would appear thus: 1+2)3 (3-6+12, &c. 3+6 -6 -6-12 12 12+24 -24 Again, divide the same, having the 2 stand first. 2+1) 3 (+3, &c. 3+3 - 3 Now let us take either quotient, with the real value of its remainder, and we shall have the same result. Thus, 3+12=15; and -6, and the remainder -24 divided by 3, gives 8, which makes-14; hence, the whole quotient. is 1. 15 Again, 3, and-3-1-7. Hence, 15-3=1, the proper quotient. 8 If we more closely examine the terms of these quotients, we shall discover that one is diverging, the other converging, and by the same ratio 2, and in general this is all a series can show, the degree of convergency. (Art. 138.) We convert quantities into series by extracting the roots of imperfect powers, as by the binomial, and by actual division, thus: Observe that these two examples are the same, except the signs of x: when that sign is plus the signs in the series will be alternately plus and minus; when minus, all will be plus. 3. What series will arise from 1+x 1-x Ans. 1+2x+2x2+2x3, &c. Observe that in this case the series commences with 2x. The unit is a proper quotient, and the series arises alone from the remainder after the quotient 1 is obtained. 2x 1-X, Observe, in this example, the term x, in the numerator does not find a place in the operation; it will be always in the remainder; therefore, a a2x2 will give the same series. 5. What series will arise from dividing 1 by 1-a+a2, or ? Ans. 1+a-a3-a1+a®+a2-a-a1, &c. In this example, observe that the signs are not alternately plus and minus, but two terms in succession plus, then two minus; this arises from there being two terms in place of one after the minus sign in the divisor. 6. What series will arise from a ? 1- -r Ans. a+ar+ar2+ar+ar1, &c. Observe that this is the regular geometrical series, as appears in (Art. 118.) 7. What series will arise from ? Ans. 1+1+1+1, &c. is 1 repeated an infinite number of times, or infinity, a result corresponding to observations under (Art. 60.) |