2 1 x 31 x2 we have x2 1 1 2x4 2x4 1 1 0 g

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Unformatted text preview: (x) = 2 1+x 1-x 1 - x2 3(1 - x2 )2 3(1 - x2 )2 and therefore g(x) is a decreasing function of x in (-1, 1), so that for all x [0, 1), g(x) g(0) = 0, or f (x) = 1+x x3 1 log x+ , 2 1-x 3(1 - x2 ) x [0, 1) . As a result, with xn = 1/(2n + 1), we get an - an+1 = = that is, 11 1 1 1 = a1 - an - an+1 - < an+1 an 12 12 12n 12(n + 1) for all n 1. 1 1 (xn )3 (xn )2 f (xn ) - 1 xn + -1 = xn xn 3(1 - (xn )2 ) 3(1 - (xn )2 ) 1 1 1 - 0, = 2-1 12n 12(n + 1) 3 (2n + 1) 2...
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