CHAPTER VII. EXAMPLES OF EXPANSION OF FUNCTIONS. 114. We shall first apply the formulæ of the preceding chapter to expand certain functions. Required the expansion of (1+x)", m not being assumed to be a positive integer. m-n f(x) = = m (m − 1) ... (m − n + 1) (1 + x)TM ̄", x)m-n−1; If x be less than 1 the last term can be made as small as we please by sufficiently increasing n, and in that case the infinite series can, by taking a sufficient number of terms, be brought as near as we please to (1 + x)m. Hence, changing a to e, and remembering that This series may be used for calculating the approximate value of e, and we may shew from it that e must be an incommensurable number. See Plane Trigonometry, Chap. x. In Arts. 115 and 116, the student will see that the last term can be made as small as we please, whatever be the value of x, if n be taken large enough. In this series, if we suppose x positive and not greater can not be greater than unity, the error we commit, if we stop at the term which does not give a very convenient form to the remainder. But by Art. 110, we may also write can be made as small as we please by taking n large enough. Hence, if n be taken large enough, the remainder can be made as small as we please. 118. In the preceding examples, we have been able to write down the general term of the series, and the remainder after n + 1 terms. But if f(x) be a complicated function, the expression for f" (x) will be generally too unwieldy for us to employ. It is, therefore, not unusual to propose such questions as "expand e log (1+x), by Maclaurin's Theorem, as far as the term involving 5." Here we are not required to ascertain the general term, or the remainder, or to shew when, for the purpose of numerical computation, the remainder may be neglected. We proceed thus |