Ex. 6. Find the limit of (x - 1)lnx when x = 1. 308. Exponential Indeterminate Forms. In order to discover under what conditions the function u" will assume an indeterminate form, as in Exs. 5 and 6, Art. 307, we write f=u", logf=v log u, the logarithms being taken in any system, and seek the conditions that make v logu indeterminate, and these will also be the conditions that make for u" indeterminate. Since log 1 = 0, log ∞ = ∞, and log 0 = − ∞, these conditions obviously are vlog u = (∞) × 0, or u" = 1±", ·(±∞) v log u = 0x (— ∞), or uo 0o. These are known as exponential indeterminate forms, and the list is evidently exhaustive for forms that result from combinations of values of u and v. But of course either u or v may by itself assume one of the algebraic forms 0/0, ∞∞, 0 × ∞, ∞ Ex. 1. The limit of (1 + x/n)" when n∞ (x = a finite quan Ex. 3. Find the limit of (ln x)x--1 when x = 1. We proceed as in Ex. 6, Art. 307, and obtain (ln x)x-1= {(ln x) ln x}(x-1)/ln x, Evaluate the limits of the following functions for the values of x indicated: 1. ln(x3-1)- ln(x-1) when x 1. х 2. ln(x2 − 1)+ ln 1 + when x 1. 3. In 2 (1 + x)} - In {3 - √(3x)} when x 3. 4. (x2 - 1) ln(x-1) when x = 1. 5. (1+1/x2) when x = ∞. 6. (1+1/x)2 when x∞. 7. x1/1nx when x 1. CHAPTER XXXI. CONVERGENCY AND DIVERGENCY OF SERIES. 309. A succession of quantities, formed in order according to some definite law, if finite in number, constitute a finite series; but if their number exceed every finite quantity, however great, they are said to form an infinite series. ... We have already seen [Art. 266] that, when › is numerically less than unity, the sum of the n terms of the geometrical progression a + ar + ar2 + ··· + am-1 can be made to differ from a/(1r) by an arbitrarily small quantity, by sufficiently increasing n. Many other series have this property, that the sum of the first n terms approaches a finite limit when n is increased ad infinitum. 310. Definitions. When the sum of a finite number of terms of an infinite series approaches a finite limit (or zero), as the number of terms is increased ad infinitum, the series is said to be convergent, and this limit is called the sum, or limit of the series. If the series have as its limit, it is said to be divergent; and if it have neither a finite nor an infinite limit, it is said to be neutral or indeterminate.* *Series of this last class are also called divergent series in many text-books of algebra. Ex. 3. The series 1-1+1 ... ad inf. is convergent, and has ad inf. has +∞ for its limit, 1+ ... ad inf. has no definite limit, but is 1 or 0 according as the number of terms is odd or even; it is therefore indeterminate. 311. It is evident that an infinite series whose terms are either all positive or all negative cannot be indeterminate, but must have either a finite or an infinite limit. If each term of an infinite series be finite (not zero) and the terms are either all positive, or all negative, the series must be divergent. For, if each term be not less than a the sum of n terms will be not less than na, and na can be made larger than any finite quantity, by sufficiently increasing n. 312. The successive terms of an infinite series will be denoted by u1, U2, U3, ・un, ..., the sum of the first n terms by Un, and the limit of the series, if it be convergent, in which case alone it has a limit, by U. Thus We shall frequently omit the words ad inf., and denote the limit of the entire series by u1+u2+ Uz + ... +un+ ... 313. In order that the series u1 + 2 + may be convergent, it is, by definition, necessary and sufficient that each of the finite sums Un, Un+1, Un×2, etc., shall approach the common limit U, as n increases. Hence Un, Un+1, Unte, etc., must differ from U, and n+2; |