3. PROP. 1. A series whose terms diminish in absolute value, and are, or end with becoming, alternately positive and negative, is convergent. Let u, – u, + Uz — U, + &c. be the proposed series or its terminal portion, the part which it follows being in the latter case supposed finite. Then, writing it in the successive forms U, — U, + (Uz — U,) + (U; – uc)+ &c. ...... (1), U, – (u, – uz) – (U, – u) - &c........ (2), and observing that uy — Uz, U, – Ug, &c. are by hypothesis positive, we see that the sum of the series is greater than U, – u, and less than u,. The series is therefore convergent. Ex. Thus the series + 114 1 1 1 1 1 - &c. ad inf. 2 3 4 5 1 tends to a limit which is less than 1 and greater than 2 4. PROP. II. A series whose nth term is of the form Un sin no (where @ is not zero or an integral multiple of 27) will converge if, for large values of n, Un retains the same sign, continually diminishes as n increases, and ultimately vanishes. Suppose un to retain its sign and to diminish continually as n increases after the term wa. Let S=u, sin að + Ua4, sin (a + 1) 0 + &c. ..... (3); * Although the above demonstration is quite rigorous, still such series present many analogies with divergent series and require careful treatment. For instance, in a convergent series where all the terms have the same sign, the order in which the terms are written does not affect the sum of the series. But in the given case, if we write the series thus, 1 + + (1+) 3+&c. 5 in which form it is equally convergent, we find that its value lies between 6 4 1 1 1 and while that of the original series lies between 1 - and 1- t i.e. 2 2' 3' 1 5 and 6° between + + Uati + &c. ... 2 sin s--, (con (a – ) - cos (a +9)o} -- foos (a + ) o-cos (a +9) =,cos (a )) + (wen - w.) cos(a +2) + (watz – Was) cos (a +ş) + &c. (a - ) Now Watz - UmW412 - 4941, &c. are all negative, hence ө 2 sin S - Ua cos 2 0 < (Ua+1 – Ua) + (Watz — Wa+1) + &c. numerically, or <U. – Wa; .. <-Wa, since Uco - 0. 0 Hence the series is convergent unless sin 2 be zero, i.e. unless O be zero or an integral multiple of 27.* An exactly similar demonstration will prove the proposition for the case in which the nth term is un sin (no – B). Ex. The series sin 20 sin 30 3 is convergent unless O be zero or a multiple of 27. This is the case although, as we shall see, the series 1 1 + &c. is divergent. + &c. 5. The theory of the convergency and divergency of series whose terms are ultimately of one sign and at the same time converge to the limit 0, will occupy the remainder of this chapter and will be developed in the following order. 1st. A fundamental proposition, due to Cauchy, which makes the test of convergency to consist in a process of integration, will be established. Žndly. Certain direct consequences of that proposition relating to particular classes of series, including the geometrical, will be deduced. 3rdly. Upon those consequences, and upon a certain extension of the algebraical theory of degree which has been developed in the writings of Professor De Morgan and of M. Bertrand, a system of criteria general in application will be founded. It may be added that the first and most important of the criteria in question, to which indeed the others are properly supplemental, being founded upon the known properties of geometrical series, might be proved without the aid of Cauchy's proposition ; but for the sake of unity it has been thought proper to exhibit the different parts of the system in their natural relation. * Malmstén (Grunert, vi. 38). A more general proposition is given by Chartier (Liouville, XVIII. 21). Fundamental Proposition. 6. PROP. III. If the function $ (2) be positive in sign but diminishing in value as a varies continuously from a to co, then the series • (a) + (a + 1) + (a + 2) + &c. ad inf. ...(4) φ (α) da is finite or infinite. For, since $ (Qc) diminishes from x=a to x=a+1, and again from x = a +1 to w = a + 2, &c., we have $*$() dx <• (a), $(x) dx < $ (a+1), and so on, ad inf. Adding these inequations together, we have 5. p(x) dx < $(a) + $ (a +1) + &c. ad inf... (3). (6). { $(x) dx > 0 (a +1) ++(a + 2) + &c. ...... Thus the integral °°(a) da, being intermediate in value between the two series $ (a) + (a + 1) + (a + 2) + &c. $ (a + 1) + $ (a + 2) + &c. which differ by $ (a), will differ from the former series by a quantity less than $ (a), therefore by a finite quantity. Thus the series and the integral are finite or infinite together. COR. If in the inequation (6) we change a into a -1, and compare the result with (5), it will appear that the series $ (a) +$ (a + 1) +$ (a + 2) + &c. ad inf. has for its inferior and superior limits 7. The application of the above proposition will be sufficiently explained in the two following examples relating to geometrical series and to the other classes of series involved in the demonstration of the final system of criteria referred to in Art. 5. Ex. 1. The geometrical series 1+h+h? +h+ &c. ad inf. is-convergent if h<1, divergent if h 1. The general term is h“, the value of x in the first term being 0, so that the test of convergency is simply whether h*dx is infinite or not. Now If h>1 this expression becomes infinite with x and the series is divergent. If h <1 the expression assumes the finite -1 value logh The series is therefore convergent. If h=1 the expression becomes indeterminate, but, proceeding in the usual way, assumes the limiting form wh* which becomes infinite with x. Here then the series is divergent. Ex. 2 The successive series 1 + 1 1 + &c. 1 1 + &c. (8)*, + &c. m a being positive, are convergent if m > 1, and divergent = The determining integrals are dx x log < (log log x)" * The convergency of these series can be investigated without the use of the Integral Calculus. See Todhunter's Algebra (Miscellaneous Theorems), or Malmstén (Grunert, VIII. 419). |