CHAPTER II CONVERGENCE 648. Definition of an Infinite Series.-Let u。, u1, u2, be any set of values, positive or negative, or both, and form the series (1) Represent the sum of the first n terms of (1) by s„: In case (ii) number. In case (i) the infinite series is said to be convergent and to have the value U, or to converge toward the value U. the infinite series is said to be divergent. is an example of a convergent series. The sum of the first n terms of the arithmetic progression 1 + 2 + 3 + 4 + n = [2a + (n−1)d] where a = 1, d = 1, n = n; U lim [" (2 + n [2635] therefore the series is divergent. Only convergent series can be used in mathematical investigations. The series " + " + "2+... ad infinitum is sometimes used instead of the limit U, or again REMARK.-The student must remember that U is not the sum of the series but the limit of the sum of the series. Similarly instead of finding "the sum of an infinite number of terms" one finds the limit of the sum of n of these terms as n increases without limit. 649. Series in which all the Terms are Positive. Examine the convergence of the series Omitting the first term of (2), compare the next n terms Disregarding the first two terms of s, and s„', each term of s,' is less than the corresponding term in s2, and hence If s is the sum of the first n terms of series (2), then the sum of its first (n + 1) terms is no matter how large n is taken. Thus it is seen that s is a variable which increases continually as n increases, but which never takes as large a value as 3. 650. Graphical Representation of These Results. Plot the successive values of s, as points on a line, 0 8-1 82 8=2.5 € 3 The preceding table shows that, as n increases by 1, the point represented by 8+ moves continuously to the right but never moves so far to the right as the point 3. Therefore, there must be some point, e, to the left of 3 (i.e.,c<3), which s„ approaches as a limit, but never reaches (2650). The table shows that the value of e correct to the third decimal place is 2.718. 651. Fundamental Theorem.-It has been shown in 650 that the variables approaches a limit as n∞; and although we do not as yet know how to calculate the numerical value of the limit e, the reasoning by which the existence of the limit e is proved is of great importance. It can be formulated as follows: If a variable sn (i) always increases when n increases, i. e., but (ii) is always less than some definite fixed number N, i. e., $n < N; for all values of n, then s, approaches a limit U; i. e., The limit U may be coincident with N or some value less than N: U≤N. EXAMPLE.-State the principle for a variable which is always decreasing and always greater than a certain fixed quantity, and draw the corresponding figure. 652. I. Comparison Test for Convergence. The following test for the convergence of an infinite series is based upon the theorem in the preceding paragraph: Let it be required to test the convergence of the infinite series all of whose terms are positive. Suppose that we can find an infinitə whose terms are all positive and which is convergent; then, if the terms of series (i) are respectively less than (or at the greatest equal to) the corresponding terms of series (ii), the series (i) is a convergent series, and its value is not greater than that of series (ii). n Then, since by hypothesis s≤S and S<N (2636), it follows that „< N and therefore, by 651, s, approaches a limit less than or at most equal to N. REMARK.-In studying the convergence of a series it is often convenient to omit a fixed number, say m, of the first terms of the series and to consider the new series thus arising. The convergence of the new series is necessary and sufficient for the convergence of the given series, for, m n +un-1) 8n-m' By hypothesis 8, is a constant, and therefore 8 will approach a limit if 8, conversely (§642). n-m does, and STANDARD SERIES FOR COMPARISON TESTS OF CONVERGENCE 653. It follows from example 10, Exercise XCI, that the series The second member is the sum of n terms and each of them is less (n + 2)p The terms in the second member form a G. P. whose ratio is 1 2p-1 which is less than 1 so long as p is greater than 1, and therefore |