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By C. WALMSLEY

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Extra info for An Introductory Course of Mathematical Analysis.

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Reversible and irreversible operations. To solve an equa- 27. tion involving an unknown, say x, we have to determine all possible values of #, if any, which satisfy the equation. In simple cases we aim at simplifying the equation by ordinary algebraical manipulation until we have reduced it to a form in which it is obvious that can only have a certain value, or one of certain several values. For example, in the case of a simple equation, by collecting on the o) left-hand side all terms containing x and on the right-hand side all terms not containing x we reduce the equation to the form t * 6, ax We whence, by division (if a={= 0)> x ^ necessarily b/a.

To INFINITE SEEIES 7] 69 question as to whether or not such a series is convergent may be We will now consider the question of the actual evaluation of the sums of such series as are convergent, partly for its own settled. g. series (i) and (v)) where a simple precise gence formula for s n can be obtained, the sum, being the limit of the can generally be obtained precisely. sequence whose nth term is sn itself. , Thus, in and, as (i), if %< 1, we saw above ((iii), p. e. >0 the as n increases in- sum of the infinite series is x).

I is even, or if n is odd, and, by taking n sufficiently great, ^! 4- . . 4- Un (or M! H- . . -f t^+i) ~2~ 2 and ^ 4- ... 4- ( or ^2 vl + ... 4 VH+I) 2" V respectively by arbitrarily little, and therefore the product differs from UV by arbitrarily little; whence it follows differ from f/" that the sum product series If now and of the product series = UV. the terms are not absolute values by u^, u 2', The ^ UV. D. (wj -f u^ -h . . series, viz. + consists of a certain set of terms, which, if replaced by their positive absolute values, would be precisely the difference between M!

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An Introductory Course of Mathematical Analysis. by C. WALMSLEY


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