9. Assuming the series for log, (1 + x) and e, shew that nearly when n is large; and find the next term of the series of which the expression on the second side is the commencement. 10. Find the coefficient of x" in the development of in which the successive terms are formed by some regular law, and the number of the terms is unlimited, is called an infinite series. 554. An infinite series is said to be convergent when the sum of the first n terms cannot numerically exceed some finite quantity however great n may be.. 555. An infinite series is said to be divergent when the sum of the first n terms can be made numerically greater than any finite quantity, by taking n large enough. 556. By the sum of an infinite series is meant the limit towards which we approximate by continually adding more and more of its terms. For example, consider the infinite series Hence if x be less than 1, however great n may be, the sum of the first n terms of the series is less than therefore convergent. And as by taking n large enough, the sum of the first n terms can be made to differ from 1 by as small 1 is the sum of the infinite series. If x= 1 the series is divergent; for the sum of the first n terms is n, and by taking sufficient terms this may be made greater than any finite quantity. If x is greater than 1 the series is divergent; for the sum 557. An infinite series in which all the terms are of the same sign is divergent if each term is greater than some assigned finite quantity, however small. For if each term is greater than the quantity e, the sum of the first ʼn terms is greater than nc, and this can be made greater than any finite quantity by taking a large enough. 558. An infinite series of terms, the signs of which are alternately positive and negative, is convergent if each term be numerically less than the preceding term. Let the series be u ̧− u ̧+u ̧− + &c.; this may be written (u ̧ − u ̧) + (μ ̧ — u ̧) + (u ̧−n ̧) + and also thus, .... u ̧ — (u,— u ̧) — (u ̧ — u ̧) — (u — u ̧) — ... From the first mode of writing the series we see that the sum of any number of terms is a positive quantity, and from the second mode of writing the series we see that the sum of any number of terms is less than u1; hence the series is convergent. It is necessary to shew in this case that the sum of any number of terms is positive; because if we only know that the sum is less than u1, we are not certain that it is not a negative quantity of unlimited magnitude. 559. An infinite series is convergent, if from and after any fixed term the ratio of each term to the preceding term is numerically less than some quantity which is itself numerically less than unity. Let the series beginning at the fixed term be and let S denote the sum of the first n of these terms. Then Now first let all the terms be positive, and suppose, Then S is less than u, {1 + k + k2 + +k"-1}; that is, less 1- k ...... Hence if k be less than unity, S is less than thus the sum of as many terms as we please beginning with u1 is less than a certain finite quantity, and therefore the series beginning with u, is convergent. Secondly, suppose the terms not all positive; then if they are all negative, the numerical value of the sum of any number of them is the same as if they were all positive; if some are positive and some negative, the sum is numerically less than if they were all positive. Hence the infinite series is still convergent. Since the infinite series beginning with u, is convergent, the infinite series which begins with any fixed term before u, will be also convergent; for we shall thus only have to add a finite number of finite terms to the series beginning with u ̧. 560. An infinite series is divergent if from and after any fixed term the ratio of each term to the preceding term is greater than unity, or equal to unity, and the terms are all of the same sign. Let the series beginning at the fixed term be and let S denote the sum of the first n of these terms. Then, Then S is numerically greater than u, {1+1+ + 1}, that is, numerically greater than nu,. Hence S may be made numerically greater than any finite quantity by taking n large enough, and therefore the series beginning with u, is divergent. T. A. 21 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. n n 562. Consider the infinite series 1 1 1 1 + + + 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 toge2 2P ther less than the following four terms are together less |