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Let us consider 1 Call the the sum sum + 2 + 22 5+ 32 i+ "" of this infinite series s of the first series (ii) above, viz. and let sn denote, as before, n terms. e. s sn ) the error after n terms* of the series, and we will denote it by En so that ] * Often called the remainder after n terms. NUMBER 70 We an estimate to the value of will try to obtain value of [CH. I En for any n. have evidently that En is the sum of the infinite series beginning with the (n+ l)th term of the original series, viz.

10. * Shew that if p a? when a having been reduced to its is negative were defined as tya* (or (^a) p), lowest terms, then, independently of the fact that INDICES 1,2] 93 the exponential function ax would not exist for some rational values of the " continuous" for any value of #; and that index, the function would not be similar difficulties would arise if the definition were modified to tya 2 is even. when q LOGARITHMS 2. 65. Definition and existence theorem for logarithms. We have seen that the equation a m = 6, in which a is any positive and m any real number, can always be solved for b in terms of a and m (b being, in fact, always positive).

Series is convergent. 2 " 2. *- soon convince us that we cannot replace the terms of sn by the corresponding terms of any geometrical progression, with common ratio less than 1, whose terms 111_ would exceed the terms of *n _l+ sn ; we could argue that + + + ... 1 + (w -_- _ i this number increases indefinitely as n increases and therefore we cannot by these means find a number such that, for all values but K of n, s n < K, However, it happens exceptionally in this case that we can actually an algebraic expression for sn thus: find 11 , 1 " 2 3 INFINITE SERIES 7] Hence, for all values of n, sn < 2, 67 and therefore the series is con- vergent.

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An Introductory Course of Mathematical Analysis. by C. WALMSLEY

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