289. The application of this principle is very important, for by means of it we can compare a given series with an auxiliary series whose convergency or divergency has been already established. The series discussed in the next article will frequently be found useful as an auxiliary series. is always divergent except when p is positive and greater than 1. 2 CASE I. Let p > 1. The first term is 1; the next two terms together are less than 4 the following four terms together are less than the fol4P 8 2P lowing eight terms together are less than 1+ + + +...; ; and so on. Hence that is, less than a geometrical progression whose common ratio is less than 1, since p>1; hence the series is convergent. 2 2P CASE II. Let p = 1. 1 1 1 1 2 3 4 5 The series now becomes 1 + + + + + The third and fourth terms together are greater than the following four terms together are greater than following eight terms together are greater than on. Hence the series is greater than 8 214 4 1 or ; 8 2 1 or the 16 2 [Art. 286.] Each term is now greater than the corresponding term in Case II., therefore the series is divergent. Hence the series is always divergent except in the case when p is positive and greater than unity. Thus if u and v denote the nth terms of the given series and the auxiliary series respectively, we have hence Lim Un-1, and therefore the two series are both convergent or both divergent. But the auxiliary series is divergent, therefore also the given series is divergent. This completes the solution of Example 1. Art. 287. 291. In the application of Art. 288 it is necessary that the limit of n should be finite; this will be the case if we find our v n auxiliary series in the following way: Take ის the nth term of the given series and retain only the highest powers of n. Denote the result by v; then the limit of is finite by Art. 270, and v, may be taken as the nth term of v the auxiliary series. Example 1. Shew that the series whose nth term is divergent. As n increases, un approximates to the value 3/2n2 $/2 1 or we have Lim Un On therefore the series whose nth term is series. But this series is divergent [Art. 290]; therefore the given series is divergent. is convergent, therefore the given series is convergent. 292. To shew that the expansion of (1+x)" by the Binomial Theorem is convergent when x < < 1. Let u,, u,+1 represent the 7th and (r+1)th terms of the expansion; then When r>n+1, this ratio is negative; that is, from this point the terms are alternately positive and negative when x is positive, and always of the same sign when x is negative. Now when r is infinite, Lim wr+1 = numerically; therefore u, since 1 the series is convergent if all the terms are of the same sign; and therefore a fortiori it is convergent when some of the terms are positive and some negative. [Art. 283.] 293. To shew that the expansion of a* in ascending powers of x is convergent for every value of x. n-1 11-1 the value of x; hence the series is convergent. 294. To shew that the expansion of log (1+x) in ascending powers of x is convergent when x is numerically less than 1. x, which in the limit is equal to x; hence the series is convergent when x is less than 1. divergent. [Art. 290.] This shews that the logarithm of zero is infinite and negative, as is otherwise evident from the equation e ̃∞ = 0. 295. The results of the two following examples are important, and will be required in the course of the present chapter. Example 1. Find the limit of log x when x is infinite. also when x is infinite y is infinite; hence the value of the fraction is zero. Example 2. Shew that when n is infinite the limit of nx"=0, when x<1. 296. It is sometimes necessary to determine whether the product of an infinite number of factors is finite or not. Suppose the product to consist of n factors and to be denoted by then if as n increases indefinitely u<1, the product will ultimately be zero, and if u,> 1 the product will be infinite; hence in order that the product may be finite, u, must tend to the limit 1. Writing 1+v for u, the product becomes (1 + v ̧) (1 + v ̧) (1 + v2) ...... (1 + v2). Denote the product by P and take logarithms; then log Plog (1+ v1) + log (1 + v2) + ... + log (1 + v) ......(1), and in order that the product may be finite this series must be convergent. since the limit of v1 is 0 when the limit of u, is 1. Hence if (2) is convergent, (1) is convergent, and the given product finite. Example. Shew that the limit, when n is infinite, of The product consists of 2n factors; denoting the successive pairs by U1, U2, uz,... and the product by P, we have therefore as in Ex. 2, Art. 291 the series is convergent, and the given product is finite. 297. In mathematical investigations infinite series occur so frequently that the necessity of determining their convergency or divergency is very important; and unless we take care that the series we use are convergent, we may be led to absurd conclusions. [See Art. 183.] |