NOTE. References to the articles in the Higher Trigonometry are given thus [Art. 100]; references to the Elementary Trigonometry thus [E. 100]. The Articles and Exercises which are marked with a star should be omitted when the subject is read for the first time. The order of the Chapters may in many cases be varied at the discretion of the teacher; in particular the last two Chapters may often be read as an Appendix to the Elementary Trigonometry. Those of the examples which are not original, have been selected from the various Examination Papers which have been set at Cambridge in the Tripos and in the different College Examinations during the past forty years. Various Examination Papers are appended for the information of intending Candidates. is of importance. Hence we prove as follows: I. Its value is less than 3. II. Since it is less than 3, the series is convergent. This may be easily calculated [See Ex. I. (1)]. For suppose that it is commensurable; it can then be put m into the form where m and n are integers. In this case n Multiply each side of this supposed identity by [n then mn-1a whole number + 1 1 + + etc. n+ 1 (n+1)(n+2) 1 is a proper fraction; for it is greater than and less than n+1 Hence we have to suppose that m❘n - 1 (a whole number) = a whole number + a proper fraction; which is absurd. V. Since the numerical value of the series is incommensurable, and we know of no surd or other algebraical expression that is equal to it, it is usual to express its numerical value by the letter e. [cf. E. 28.] EXAMPLES. I. (1) Calculate the value of e by taking the first 13 terms of the series. (2) Prove that the first 13 terms of the series will give the value of e correct to 9 significant figures. (6) Prove that the series x- ‡x2+}x3 − {x2+etc. is convergent if x is greater than -1 and is not greater than 1. |