PureMath.pdf

# X 1 x 2 x 3 2 x 1 x 2 x 3 x 4 math trip 1908 the

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( x + 1)( x + 2)( x + 3) + 2! ( x + 1)( x + 2)( x + 3)( x + 4) + . . . . ( Math. Trip. 1908.) [The difference between 1 / ( x + 1) 2 and the sum of the first n terms of the series is 1 ( x + 1) 2 n ! ( x + 2)( x + 3) . . . ( x + n + 1) . ] 28. No equation of the type Ae αx + Be βx + · · · = 0 , where A , B , . . . are polynomials and α , β , . . . different real numbers, can hold for all values of x . [If α is the algebraically greatest of α , β , . . . , then the term Ae αx outweighs all the rest as x → ∞ .] 29. Show that the sequence a 1 = e, a 2 = e e 2 , a 3 = e e e 3 , . . . tends to infinity more rapidly than any member of the exponential scale. [Let e 1 ( x ) = e x , e 2 ( x ) = e e 1 ( x ) , and so on. Then, if e k ( x ) is any member of the exponential scale, a n > e k ( n ) when n > k .] 30. Prove that d dx { φ ( x ) } ψ ( x ) = d dx { φ ( x ) } α + d dx { β ψ ( x ) }

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[IX : 216] THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS 489 where α is to be put equal to ψ ( x ) and β to φ ( x ) after differentiation. Establish a similar rule for the differentiation of φ ( x ) [ { ψ ( x ) } χ ( x ) ] . 31. Prove that if D n x e - x 2 = e - x 2 φ n ( x ) then (i) φ n ( x ) is a polynomial of degree n , (ii) φ n +1 = - 2 n + φ 0 n , and (iii) all the roots of φ n = 0 are real and distinct, and separated by those of φ n - 1 = 0. [To prove (iii) assume the truth of the result for κ = 1, 2, . . . , n , and consider the signs of φ n +1 for the n values of x for which φ n = 0 and for large (positive or negative) values of x .] 32. The general solution of f ( xy ) = f ( x ) f ( y ), where f is a differentiable function, is x a , where a is a constant: and that of f ( x + y ) + f ( x - y ) = 2 f ( x ) f ( y ) is cosh ax or cos ax , according as f 00 (0) is positive or negative. [In proving the second result assume that f has derivatives of the first three orders. Then 2 f ( x ) + y 2 { f 00 ( x ) + y } = 2 f ( x )[ f (0) + yf 0 (0) + 1 2 y 2 { f 00 (0) + 0 y } ] , where y and 0 y tend to zero with y . It follows that f (0) = 1, f 0 (0) = 0, f 00 ( x ) = f 00 (0) f ( x ), so that a = p f 00 (0) or a = p - f 00 (0).] 33. How do the functions x sin(1 /x ) , x sin 2 (1 /x ) , x cosec(1 /x ) behave as x +0? 34. Trace the curves y = tan xe tan x , y = sin x log tan 1 2 x . 35. The equation e x = ax + b has one real root if a < 0 or a = 0, b > 0. If a > 0 then it has two real roots or none, according as a log a > b - a or a log a < b - a . 36. Show by graphical considerations that the equation e x = ax 2 + 2 bx + c has one, two, or three real roots if a > 0, none, one, or two if a < 0; and show how to distinguish between the different cases. 37. Trace the curve y = 1 x log e x - 1 x , showing that the point (0 , 1 2 ) is a centre of symmetry, and that as x increases through all real values, y steadily increases from 0 to 1. Deduce that the equation 1 x log e x - 1 x = α has no real root unless 0 < α < 1, and then one, whose sign is the same as that of α - 1 2 . [In the first place y - 1 2 = 1 x log e x - 1 x - log e 1 2 x = 1 x log sinh 1 2 x 1 2 x !
[IX : 216] THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS 490 is clearly an odd function of x . Also dy dx = 1 x 2 ( 1 2 x coth 1 2 x - 1 - log sinh 1 2 x 1 2 x !) . The function inside the large bracket tends to zero as x 0; and its derivative is 1 x 1 - 1 2 x sinh 1 2 x !

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