5 employ the exponential series to prove that e x

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5. Employ the exponential series to prove that e x tends to infinity more rapidly than any power of x . [Use the inequality e x > x n /n !.] 6. Show that e is not a rational number. [If e = p/q , where p and q are integers, we must have p q = 1 + 1 + 1 2! + 1 3! + · · · + 1 q ! + . . . or, multiplying up by q !, q ! p q - 1 - 1 - 1 2! - · · · - 1 q ! = 1 q + 1 + 1 ( q + 1)( q + 2) + . . . and this is absurd, since the left-hand side is integral, and the right-hand side less than { 1 / ( q + 1) } + { 1 / ( q + 1) } 2 + · · · = 1 /q .]
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[IX : 212] THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS 474 7. Sum the series 0 P r ( n ) x n n ! , where P r ( n ) is a polynomial of degree r in n . [We can express P r ( n ) in the form A 0 + A 1 n + A 2 n ( n - 1) + · · · + A r n ( n - 1) . . . ( n - r + 1) , and X 0 P r ( n ) x n n ! = A 0 X 0 x n n ! + A 1 X 1 x n ( n - 1)! + · · · + A r X r x n ( n - r )! = ( A 0 + A 1 x + A 2 x 2 + · · · + A r x r ) e x . ] 8. Show that X 1 n 3 n ! x n = ( x + 3 x 2 + x 3 ) e x , X 1 n 4 n ! x n = ( x + 7 x 2 + 6 x 3 + x 4 ) e x ; and that if S n = 1 3 + 2 3 + · · · + n 3 then X 1 S n x n n ! = 1 4 (4 x + 14 x 2 + 8 x 3 + x 4 ) e x . In particular the last series is equal to zero when x = - 2. ( Math. Trip. 1904.) 9. Prove that ( n/n !) = e , ( n 2 /n !) = 2 e , ( n 3 /n !) = 5 e , and that ( n k /n !), where k is any positive integer, is a positive integral multiple of e . 10. Prove that 1 ( n - 1) x n ( n + 2) n ! = ( x 2 - 3 x + 3) e x + 1 2 x 2 - 3 /x 2 . [Multiply numerator and denominator by n + 1, and proceed as in Ex. 7.] 11. Determine a , b , c so that { ( x + a ) e x + ( bx + c ) } /x 3 tends to a limit as x 0, evaluate the limit, and draw the graph of the function e x + bx + c x + a . 12. Draw the graphs of 1 + x , 1 + x + 1 2 x 2 , 1 + x + 1 2 x 2 + 1 6 x 3 , and compare them with that of e x . 13. Prove that e - x - 1 + x - x n 2! + · · · - ( - 1) n x n n ! is positive or negative according as n is odd or even. Deduce the exponential theorem. 14. If X 0 = e x , X 1 = e x - 1 , X 2 = e x - 1 - x, X 3 = e x - 1 - x - ( x 2 / 2!) , . . . ,
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[IX : 213] THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS 475 then dX ν /dx = X ν - 1 . Hence prove that if t > 0 then X 1 ( t ) = Z t 0 X 0 dx < te t , X 2 ( t ) = Z t 0 X 1 dx < Z t 0 xe x dx < e t Z t 0 x dx = t 2 2! e t , and generally X ν ( t ) < t ν ν ! e t . Deduce the exponential theorem. 15. Show that the expansion in powers of p of the positive root of x 2+ p = a 2 begins with the terms a { 1 - 1 2 p log a + 1 8 p 2 log a (2 + log a ) } . ( Math. Trip. 1909.) 213. The logarithmic series. Another very important expansion in powers of x is that for log(1 + x ). Since log(1 + x ) = Z x 0 dt 1 + t , and 1 / (1 + t ) = 1 - t + t 2 - . . . if t is numerically less than unity, it is natural to expect * that log(1 + x ) will be equal, when - 1 < x < 1, to the series obtained by integrating each term of the series 1 - t + t 2 - . . . from t = 0 to t = x , i.e. to the series x - 1 2 x 2 + 1 3 x 3 - . . . . And this is in fact the case. For 1 / (1 + t ) = 1 - t + t 2 - · · · + ( - 1) m - 1 t m - 1 + ( - 1) m t m 1 + t , and so, if x > - 1, log(1 + x ) = Z x 0 dt 1 + t = x - x 2 2 + · · · + ( - 1) m - 1 x m m + ( - 1) m R m , where R m = Z x 0 t m dt 1 + t . * See Appendix II for some further remarks on this subject.
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[IX : 214] THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS 476 We require to show that the limit of R m , when m tends to , is zero.
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