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# 1 1 2 4 x n 2 2 which is at any rate fairly rapidly

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1 · 1 2 · 4 x N 2 2 ± . . . , which is at any rate fairly rapidly convergent, and may be very rapidly so. Thus 67 = 64 + 3 = 8 ( 1 + 1 2 3 64 - 1 · 1 2 · 4 3 64 2 + . . . ) . Let us consider the error committed in taking 8 3 16 (the value given by the first two terms) as an approximate value. After the second term the terms alternate in sign and decrease. Hence the error is one of excess, and is less than 3 2 / 64 2 , which is less than . 003. 3. If x is small compared with N 2 then p N 2 + x = N + x 4 N + Nx 2(2 N 2 + x ) , the error being of the order x 4 /N 7 . Apply the process to 907. [Expanding by the binomial theorem, we have p N 2 + x = N + x 2 N - x 2 8 N 3 + x 3 16 N 5 , the error being less than the numerical value of the next term, viz. 5 x 4 / 128 N 7 . Also Nx 2(2 N 2 + x ) = x 4 N 1 + x 2 N 2 - 1 = x 4 N - x 2 8 N 3 + x 3 16 N 5 , the error being less than x 4 / 32 N 7 . The result follows. The same method may be applied to surds other than quadratic surds, e.g. to 3 1031.] 4. If M differs from N 3 by less than 1 per cent. of either then 3 M differs from 2 3 N + 1 3 ( M/N 2 ) by less than N/ 90 , 000. ( Math. Trip. 1882.)

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[IX : 216] THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS 482 5. If M = N 4 + x , and x is small compared with N , then a good approxi- mation for 4 M is 51 56 N + 5 56 M N 3 + 27 Nx 14(7 M + 5 N 4 ) . Show that when N = 10, x = 1, this approximation is accurate to 16 places of decimals. ( Math. Trip. 1886.) 6. Show how to sum the series X 0 P r ( n ) m n x n , where P r ( n ) is a polynomial of degree r in n . [Express P r ( n ) in the form A 0 + A 1 n + A 2 n ( n - 1) + . . . as in Ex. xc . 7.] 7. Sum the series 0 n m n x n , 0 n 2 m n x n and prove that X 0 n 3 m n x n = { m 3 x 3 + m (3 m - 1) x 2 + mx } (1 + x ) m - 3 . 216. An alternative method of development of the theory of the exponential and logarithmic functions. We shall now give an outline of a method of investigation of the properties of e x and log x entirely different in logical order from that followed in the preceding pages. This method starts from the exponential series 1 + x + x 2 2! + . . . . We know that this series is convergent for all values of x , and we may therefore define the function exp x by the equation exp x = 1 + x + x 2 2! + . . . . (1) We then prove, as in Ex. lxxxi . 7, that exp x × exp y = exp( x + y ) . (2) Again exp h - 1 h = 1 + h 2! + h 2 3! + · · · = 1 + ρ ( h ) ,
[IX : 216] THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS 483 where ρ ( h ) is numerically less than | 1 2 h | + | 1 2 h | 2 + | 1 2 h | 3 + · · · = | 1 2 h | / (1 - | 1 2 h | ) , so that ρ ( h ) 0 as h 0. And so exp( x + h ) - exp x h = exp x exp h - 1 h exp x as h 0, or D x exp x = exp x. (3) Incidentally we have proved that exp x is a continuous function. We have now a choice of procedure. Writing y = exp x and observing that exp 0 = 1, we have dy dx = y, x = Z y 1 dt t , and, if we define the logarithmic function as the function inverse to the expo- nential function, we are brought back to the point of view adopted earlier in this chapter.

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