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 4 of 6: Isolate four terms and show the rest is positive

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d(x)=\underbrace{\frac1{x-1}-\frac1x-\frac1{x-2}+\frac1{x-3}}_{\text{four dominant terms}}+\sum_{k=2}^{1008}\Big(\frac1{x-(2k+1)}-\frac1{x-2k}\Big)-\frac1{x-2018}

Detailed analysis

For $1<x<2$ and $k=2,\dots,1008$ , both $x-(2k+1)$ and $x-2k$ are negative with $x-(2k+1)<x-2k$ , so $\frac1{x-(2k+1)}>\frac1{x-2k}$ and each bracketed term in the sum is positive. Also $x-2018<0$ so $-\frac1{x-2018}>0$ . Hence the sum of all terms beyond the first four is strictly positive, and it suffices to prove that the four dominant terms alone exceed $2$ .