Simplify inequalities first. Right:

sum_{i= 1}^n i/(4i^2- 1)

=[sum_{i= 1}^n 1/(2i- 1)]/2-[n/(2n+ 1)]

& gt[sum_{i= 1}^n 1/(2i- 1)]/2- 1/4。 ( 1)

Then estimate the sum in (1): because n >；; =3, so

sum_{i= 1}^n 1/(2i- 1)

= 1+ 1/3+sum_{i=3}^n 1/(2i- 1)

& gt 1+ 1/3+int_{i=2}^n 1/(2i+ 1)

= 4/3 + [ln((2n+ 1)/5)]/2。 (2)

Substitute (2) into (1), and get the lower bound estimate on the right side of the original inequality:

R & gt5/ 12+[ln((2n+ 1)/5)]/4。 (3)

On the left of the original inequality is

L = [ln(2n+ 1)]/4。 (4)

As can be seen from (3) and (4), L; ln(5)。

According to the manual test, the left side of the formula is greater than 1.66 and the right side is less than 1.5438+0.