their sum remains less than a fixed number N no matter how large ris (2540, IV). Hence, whatever r is, we have, Sr-SN and .. 2 As r increases indefinitely, Sr increases indefinitely and approaches a limit. The case when increases indefinitely according to any law whatever is reducible to the preceding case. It is sufficient to consider the highest positive integral power of 2 which is contained in n; if is the exponent of this power when Sqr <S' n < Sqr+1 increases indefinitely, n in general not integral, increases indefinitely and Sr and Sr+1 approach a common limit, and S, which is comprised between them, approaches the same limit. convergent when p> 1. Therefore the given series is 654. Series (3) is useful as a test series, for many series which can not be proved convergent by means of the geometric series can be proved convergent by using (3). For example, The harmonic series is derived from series (3) by putting p = 1. If S is the sum of the first n terms, then There are n terms in the second member, the smallest of which is hence, by adding these inequalities, we get Sqr — S2> - 1 2 2 Therefore, since n, and consequently m, increases without limit, r increases without limit; hence Sr increases without limit. But if we take n > 2', we have Sn> Sqr. Therefore S increases without limit when n increases indefinitely, and the series is divergent. 656. Test for Divergence. The test for the divergence of a series may be established in the same way as the test for convergence was derived in 648-650, thus: be a series of positive terms which is to be tested for divergence. can be found such that every term of (i) is greater than or equal to the corresponding term of (ii), then series (i) is divergent. which is known to be divergent, and therefore, according to the previous test, series (i) is divergent. 657. II. Ratio Test For Convergence.-Let it be required to test the series for convergence. Form the test ratio Un+1 This ratio will in Un general approach a fixed limit or increase without limit when n is indefinitely increased. If the ratio has a limit let the limit be r. 658. THEOREM.-If r<1, the series is convergent; if r>1, it is divergent; if r = 1 the series can not be said to be convergent or divergent without further examination. Un +1 Un Case I. r<1. Then, as n increases, the points corresponding to the value of will arrange themselves about the point r, and hence if a fixed point p is chosen at will between r and 1, the point Un+1 will, in case n is taken large enough (i. e., for n equal to or greater than a certain fixed number m), lie to the left of Р and we Un The sum of the terms of the series beginning with um+1 can never however large t is, i. e., however many terms p are taken. Therefore the u series is convergent. Case II. divergent. r = 1. The series can not be said to be convergent or For consider the series (3). Then Thenr 1, no matter what p is. But when p> 1, (3) converges (2653); and when p < 1, (3) diverges (2656, Ex. 2). That is, r may equal 1 both for a convergent and for a divergent series. NOTE. The student should note that the theorem requires that the limit of the ratio Un+1 un series, is always less than 1, in case the series is convergent. Thus, in case of the harmonic n+1 Here the ratio is less than 1 for all values of n, yet the series is divergent ($655); un but the limit of the ratio is not less than 1 but equal to it. Therefore the series is convergent for all finite values of x. is convergent if n is greater than 2, divergent if n is less than or equal to 2. . each 14. Suppose that in the series u +, +2 + 3 + term is less than the preceding; then show that this series and the series +2μ2+ 22 μ ̧ + 23 μ, + 2* u 15 0 3 convergent or both divergent. + SERIES WITH POSITIVE AND NEGATIVE TERMS are both 659. Alternating Series.-THEOREM.-Suppose that the terms of the given series are alternately positive and negative, and that each term is less than or equal to the one which precedes it, Throughout the steps of the proof which is to follow, consider as an example the series Outline of the Plan of the Proof of the Theorem |