Let f(x) and g(x) be given by f(x)= (1/x) +(1/(x−2)) +(1/(x−4)) + ... +(1/(x−2018)) and g(x)=(1/(x−1)) +(1/(x−3)) +(1/(x−5)) +...+ (1/(x−2017)). Prove that ∣ f(x)−g(x)∣ >2 for any non−integer real number x satisfying 0<x<2018.


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Question Number 201441 by dimentri last updated on 06/Dec/23

$L e t f (x) a n d g (x) b e g i v e n b y$ $f (x) = \frac{1}{x} + \frac{1}{x - 2} + \frac{1}{x - 4} + . . . + \frac{1}{x - 2018}$ $a n d$ $g (x) = \frac{1}{x - 1} + \frac{1}{x - 3} + \frac{1}{x - 5} + . . . + \frac{1}{x - 2017} .$ $P r o v e t h a t ∣ f (x) - g (x) ∣ > 2$ $f o r a n y n o n - i n t e g e r r e a l n u m b e r$ $x s a t i s f y i n g 0 < x < 2018 .$
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