A different type of converging sum

A lower bound is

$\underline{S}= \zeta(2)+\zeta(3)+\zeta(5)+\zeta(7)-\zeta(6)-\zeta(10)-\zeta(14)-\zeta(15)-\zeta(21)-\zeta(35)+\zeta(30)+\zeta(42)+\zeta(70)+\zeta(105)-\zeta(210)-1 \approx 0.87383805$

while an upper bound is

$\bar{S} = 1 - \frac{1}{3\times4} - \frac{1}{7\times8}-\frac{1}{8\times9}-\frac{1}{15\times16}-\frac{1}{24\times25}-\frac{1}{26\times27}-\frac{1}{32\times32}-\frac{1}{35\times36}-\frac{1}{48\times49}-\frac{1}{63\times64}-\frac{1}{80\times81}-\frac{1}{99\times100}\approx 0.87483257$

This is because

$\sum_{k=2}^\infty \frac{1}{n^k} = \frac{\frac{1}{n^2}}{1-\frac{1}{n}} = \frac{1}{n(n-1)}=\frac{1}{n-1}-\frac{1}{n}$

$\sum_{n=2}^\infty [\frac{1}{n-1}-\frac{1}{n}] = 1$

while some of the terms are duplicates and can be removed.

Therefore, $S\approx 0.87$ up to 2 decimal places.