Next, suppose the ratio of each term to the preceding to be unity; then S = nu1, and this may be made greater than any finite quantity by taking n large enough. And if we begin with any fixed term before u, the series will obviously still be divergent. 561. The rules in the preceding articles will determine in many cases whether an infinite series is convergent or divergent. There is one case in which they do not apply which it is desirable to notice, namely, when the ratio of each term to the preceding is less than unity, but continually approaching unity, so that we cannot name any finite quantity k which is less than unity, and yet always greater than this ratio. In such a case, as will appear from the example in the following article, the series may be convergent or divergent. 562. Consider the infinite series Here the ratio of the nth term to the (n-1)th term is -1); if p be positive, this is less than unity, but continually approaches to unity as n increases. This case then cannot be tested by any of the rules already given; we shall however prove that the series is convergent if p be greater than unity, and divergent if p be unity, or less than unity. I. Suppose p greater than unity. The first term of the series is 1, the next two terms are toge 2 ther less than the following four terms are together less 4 4P' 2P' the following eight terms are together less than 8 than and so on. Hence the whole series is less than 2 4 1+ + 8 + 2 where x = 2P Since p is greater than unity, x is less than unity; hence the series is convergent. II. Suppose p equal to unity. 2 1 2' are together greater than or the following four terms are 4 1 together greater than or 8 2' and so on. Hence by taking a sufficient number of terms we can obtain a sum greater than 1 any finite multiple of; the series is therefore divergent. III. Suppose p less than unity or negative. Each term is now greater than the corresponding term in II.; the series is therefore a fortiori divergent. 563. We will now give a general theorem which can be proved in the manner exemplified in the preceding article. If (x) be positive for all positive integral values of x, and continually diminish as x increases, and m be any positive integer, then the two infinite series and ☀ (1) + $ (2) + $ (3) + ☀ (4) + ☀ (5) + $ (1) +mp (m) + m2 4 (m3) + m3¿ (m3) + ...... are both convergent or both divergent. Consider all the terms of the first series comprised between (m2) and (m**1), including the last and excluding the first, k being any positive integer; the number of these terms is m*+1-m", and their sum is therefore greater than m (m-1) (m+1). Thus all the first series beginning with the term (m+1) will be times the second series beginning with the Thus if the second series be divergent, so also Again, the terms selected from the first series are less than m* (m − 1) $ (m2). Thus all the first series beginning with the term (m+1) will be less than m-1 times the second series beginning with mp (m). Thus if the second series be convergent, so also is the first. As an example of the use of this theorem we may take the 1 following; the series of which the general term is n (log n)P is con vergent if p be greater than unity, and divergent if p be equal to unity or less than unity. By the theorem the proposed series is convergent or divergent according as the series of which the 564. The series obtained by expanding (1 + x)" by the binomial theorem is convergent if x be less than unity. For the ratio of the (r+ 1)th term to the 7th is when r is greater than n, the factor n―r+1 is numerically less than unity, though it continually approaches to unity. If then be less than unity, the product n−r+1 x will, when r is greater than n, be always numerically less than a quantity which is itself numerically less than unity. Hence the series is convergent. (Art. 559.) 565. The series obtained by expanding log (1+x) in powers of x is convergent if x be less than unity. rx For the ratio of the (~+1)th term to the this. If then r+1 x be less than unity, this ratio is always numerically less than a quantity which is itself numerically less than unity. Hence the series is convergent. (Art. 559.) 566. The series obtained by expanding a in powers of is always convergent. x log, a For the ratio of the (r+1)th term to the 7th is What r ever be the value of x, we can take r so large that this ratio shall be less than unity, and the ratio will diminish as r increases. Hence the series is always convergent. (Art. 559.) EXAMPLES OF CONVERGENCY AND DIVERGENCY OF SERIES. Examine whether the following ten series are convergent or divergent : 11. Suppose that in the series u+u2+U2+U ̧+ 1 3 is less than the preceding; then shew that this series and the series is convergent if n be greater than 2, and divergent if ʼn be less than 2 or equal to 2. XLI. INTEREST. 567. Interest is money paid for the use of money. The sum lent is called the Principal. The Amount is the sum of the Principal and Interest at the end of any time. 568. Interest is of two kinds, simple and compound. When interest of the Principal alone is taken it is called simple interest; but if the interest as soon as it becomes due is added to the principal and interest charged upon the whole, it is called compound interest. 569. The rate of interest is the money paid for the use of a certain sum for a certain time. In practice the sum is usually £100 and the time one year; and when we say that the rate of interest is £4. 6s. 8d. per cent., we mean that £4. 6s. 8d., that is, £43, is due for the use of £100 for one year. In theory it is convenient, as we shall see, to use a symbol to denote the interest of one pound for one year. |