14 MATHEMATICAL THEORY OF INVESTMENT Again, suppose it be required to find the value of lim"2-9^14. ^=2 x'-llx+lS The expression has no sense when x = 2, for then it takes the form ^, but both numerator and denominator of the fraction are divisible by x — 2, so that y_9^+14 2.-7 5 i^-ii^+is^^roT EXAMPLES 1. Prove that lim ;„ = lim ——m— n=oo n=co 1 — T a
1-r 2. Find the value of lim sa ~ 5 x + 6. a;=3 X — 3 3. Find the value of ^ C(^-+ ^)2-5(^ + A) + 61-^-5^+61 )i=o , A 4. Find the value of lim (x + ^" ~ xn. h=0 ft 5. Find the value of lim33"6 + 6xi- 7 xs +17. »=« 11 a6 + 29 z" - 43 6. Find the value of lim 6X7 + 3XS-1^ + 20. ^=» 14 .i-6 - 7 a:5 + 347 7. Find the value of lim 29 xl - 3 xe + u Ja. ^=« 31 a-8+16 8. What is the value of lim "f" + a^""1 + + am in the following cases? a-" V + 6^"-1 + + /'„ " (a) m > n, (b) m = n, (c) m < »i. 5. Infinite series with constant terms. Consider the infinite sequence of constant terms MI, ^, Mg, , M,, . The expression ^ + u^ + u^+ + u^+ LIMITS AND SERIES- .—--— l5 is called an infinite series. Examples of infinite series have already been found in the section on infinite geometrical progressions. Other examples would be (1) 1+2+3+4+.--. (2)1+|+|+^+-- C^-H-^--- <4)1+^+112+^J!-3+"" In the case of the infinite geometrical progressions the series were of value because it was possible to find in each case a fixed number which was the limit of the sum toward which the sum of the first n terms of the series was approaching as n was in- creased indefinitely. To generalize this notion, consider the sum s^=u^+u^+ +u^ formed by taking the first n terms of the infinite series. If the number of terms in «„ is increased indefinitely, one of three things will be true: (1) The snm »„ approaches a limit which is finite and of course definite; fo"r example, the limit of the sum of the first n terms of the series -, -i l+-2+¥+ of the stick problem is 2. (2) The sum »„ will increase without limit, as in the case of the series i+^+^+.... To see that the sum of n terms of this series will increase without limit, we have only to note that the terms after the second may be grouped by taking the third and fourth, the fifth to the eighth, the ninth to the sixteenth, and so on, and in each case the sum of the terms in a group is greater than y The process of grouping may be continued indefinitely, so that the series will appear as the sum of an infinite number of finite terms.