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-gt-2-for-any-non-in

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} + \dots + \frac{1}{x - 2018}

a n d

g (x) = \frac{1}{x - 1} + \frac{1}{x - 3} + \frac{1}{x - 5} + \dots + \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 .

Answered by Rasheed.Sindhi last updated on 06/Dec/23

∣ f (x) - g (x) ∣=

∣ \frac{{(- 1)}^{1 + 1}}{x + 1 - 1} + \frac{{(- 1)}^{2 + 1}}{x + 1 - 2} + \frac{{(- 1)}^{3 + 1}}{x + 1 - 3} + \dots + \frac{{(- 1)}^{2018 + 1}}{x + 1 - 2018} + \frac{{(- 1)}^{2019 + 1}}{x + 1 - 2019} ∣

=∣ \underset{n = 1}{\overset{2019}{Σ}} (\frac{{(- 1)}^{n + 1}}{x - n + 1}) ∣

\dots .