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S'. , , , Furthermore, every number s2n+i of the sequence (7 a) ^ every number 52m of the sequence (7 6), for - szrn = (^2m+i - ^1+2) 4- ... 4- (lln-i ~ uzn) + U'm+i if W ^ m, *an+i = (Wgn+a - Or 2n+3 ) 4- ... 4- (ttam-a i< - '^2m-i) 4- W m 2 if W < m. Therefore the lower bound S of the first sequence (7 a), which exceeds (or equals) every number less than all the numbers s lt ss of the sequence (7 a), must exceed (or equal) every number of the second sequence (76); and therefore 8 exceeds, or is equal to, the , upper bound 8' of this sequence.

Loga , We , , ^? ), Q. E. D. ft. notice further that, whatever the base (a\ loga 1 = 0, because = l. a Thus we know that if the base (a) is greater than 1, every positive number b has a logarithm, which is positive or negative according as the number is greater than or less than 1, and that the logarithm increases steadily (and continuously) as the number increases. If the base is less than 1, the logarithm is number is less or greater than the number increases. as the as * The modifications when a < 1 positive or negative according 1 and the logarithm decreases are evident.

Which is 14 than less sum Evidently the is , less of the series can be found correct to any desired degree of accuracy simply by ascertaining what value of n will suffice to make n less than the assigned degree of error and adding up the first n terms of the series. E The student may have noticed that in passing from the ex- pression 1 L + Jn 2 n n~+ JL ^ '" + 1 2 n+1 L_ n+m l 2' H m " _+_. + ^ n+m to the expression we have made a very generous allowance; not only is the first of these expressions less than the second, but it is very considerably less, being in fact, almost as evidently, less than one nth of can indeed assert that it.

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An Introductory Course of Mathematical Analysis. by C. WALMSLEY

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