Problem 2

Let

f(x)

and

g(x)

be given by

f(x)=\dfrac1x+\dfrac1{x-2}+\dfrac1{x-4}+\cdots+\dfrac1{x-2018}

and

g(x)=\dfrac1{x-1}+\dfrac1{x-3}+\dfrac1{x-5}+\cdots+\dfrac1{x-2017}

. Prove that

|f(x)-g(x)|>2

for any non-integer real number

x

satisfying

0<x<2018

Step 5 of 6: Regroup the four terms

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\frac1{x-1}-\frac1x-\frac1{x-2}+\frac1{x-3}=\Big(\frac1{x-1}+\frac1{2-x}\Big)+\frac3{x(x-3)}

Detailed analysis

Since $-\frac1{x-2}=\frac1{2-x}$ , and $-\frac1x+\frac1{x-3}=\frac{-(x-3)+x}{x(x-3)}=\frac3{x(x-3)}$ , the four dominant terms split exactly into the displayed sum of a symmetric pair and a single fraction.