V v 1 u v u u 1 u 2 v v 1 v 2 u u 1 v v 1 and the sum

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v 0 + v 1 ) - u 0 v 0 , ( u 0 + u 1 + u 2 )( v 0 + v 1 + v 2 ) - ( u 0 + u 1 )( v 0 + v 1 ) , . . . and the sum of the first n + 1 groups is ( u 0 + u 1 + · · · + u n )( v 0 + v 1 + · · · + v n ) , and tends to st as n → ∞ . When the sum of the series is formed in this manner the sum of the first one, two, three, . . . groups comprises all the terms in the first, second, third, . . . rectangles indicated in the diagram above. The sum of the series formed in the second manner is st . But the first series is (when the brackets are removed) a rearrangement of the second; and therefore, by Dirichlet’s Theorem, it converges to the sum st . Thus the theorem is proved.
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[VIII : 171] THE CONVERGENCE OF INFINITE SERIES, ETC. 390 Examples LXVIII. 1. Verify that if r < 1 then 1 + r 2 + r + r 4 + r 6 + r 3 + · · · = 1 + r + r 3 + r 2 + r 5 + r 7 + · · · = 1 / (1 - r ) . 2. * If either of the series u 0 + u 1 + . . . , v 0 + v 1 + . . . is divergent, then so is the series u 0 v 0 + ( u 1 v 0 + u 0 v 1 ) + ( u 2 v 0 + u 1 v 1 + u 0 v 2 ) + . . . , except in the trivial case in which every term of one series is zero. 3. If the series u 0 + u 1 + . . . , v 0 + v 1 + . . . , w 0 + w 1 + . . . converge to sums r , s , t , then the series λ k , where λ k = u m v n w p , the summation being extended to all sets of values of m , n , p such that m + n + p = k , converges to the sum rst . 4. If u n and v n converge to sums s and t , then the series w n , where w n = u l v m , the summation extending to all pairs l , m for which lm = n , converges to the sum st . 171. Further tests for convergence and divergence. The exam- ples on pp. 385 387 suffice to show that there are simple and interesting types of series of positive terms which cannot be dealt with by the general tests of § 168 . In fact, if we consider the simplest type of series, in which u n +1 /u n tends to a limit as n → ∞ , the tests of § 168 generally fail when this limit is 1. Thus in Ex. lxvii . 5 these tests failed, and we had to fall back upon a special device, which was in essence that of using the series of Ex. lxvii . 4 as our comparison series, instead of the geometric series. The fact is that the geometric series, by comparison with which the tests of § 168 were obtained, is not only convergent but very rapidly convergent, far more rapidly than is necessary in order to ensure convergence. The tests derived from comparison with it are therefore naturally very crude, and much more delicate tests are often wanted. We proved in Ex. xxvii . 7 that n k r n 0 as n → ∞ , provided r < 1, whatever value k may have; and in Ex. lxvii . 1 we proved more than this, viz. that the series n k r n is convergent. It follows that the sequence r , r 2 , r 3 , . . . , r n , . . . , where r < 1, diminishes more rapidly than the sequence 1 - k , 2 - k , 3 - k , . . . , n - k , . . . . This seems at first paradoxical if r is not much less than unity, and k is large. Thus of the two sequences 2 3 , 4 9 , 8 27 , . . . ; 1 , 1 4096 , 1 531 , 441 , . . .
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[VIII : 172] THE CONVERGENCE OF INFINITE SERIES, ETC. 391 whose general terms are ( 2 3 ) n and n - 12 , the second seems at first sight to decrease far more rapidly. But this is far from being the case; if only we go far enough into the sequences we shall find the terms of the first sequence very much the smaller. For example, (2 / 3) 4 = 16 / 81 < 1 / 5 , (2 / 3) 12 < (1 / 5) 3 < (1 / 10) 2 , (2 / 3) 1000 < (1 / 10) 166 , while 1000 - 12 = 10 - 36 ;
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