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 1 of 6: Write f and g as sums

Previous step Next step

f(x)=\sum_{k=0}^{1009}\frac1{x-2k},\qquad g(x)=\sum_{k=0}^{1008}\frac1{x-(2k+1)}

Detailed analysis

The function $f$ sums $1010$ terms with even shifts $0,2,\dots,2018$ , and $g$ sums $1009$ terms with odd shifts $1,3,\dots,2017$ . Both are undefined exactly at the integers $0,1,\dots,2018$ , which are excluded by hypothesis.