Can a fraction in a sum equation be flipped?

Question

Given the equation:

$$\sum_i^n \frac{1}{f(i)} \geq \sum_i^n \frac{1}{g(i)} $$

It looks intuitive to be able to flip the fraction and the sign, such that:

$$\sum_i^n f(i) \leq \sum_i^n g(i)$$

Is this true? If so, how to prove it?

$1+4 \ge 2+3$ and $1 + 1/4 > 1/2 + 1/3$. How many examples did you try to test your intuition? — Martin R, Apr 12 '23 at 19:39

score 2 · Accepted Answer · answered Apr 12 '23 at 19:35

2

What about $n=2$ and $g(1)=g(2)=2$, $f(1)=1$ and $f(2)=1000$?

answered Apr 12 '23 at 19:35

1 Answers1