Bài giải:

H = $2+\frac{2}{3}+\frac{2}{9}+\frac{2}{27}+...+\frac{2}{729}+\frac{2}{2187}$

3 x H = $6+2+\frac{2}{3}+\frac{2}{9}+\frac{2}{27}+...+\frac{2}{729}$

3 x H – H = $\left( 6+2+\frac{2}{3}+\frac{2}{9}+\frac{2}{27}+...+\frac{2}{729} \right)-\left( 2+\frac{2}{3}+\frac{2}{9}+\frac{2}{27}+...+\frac{2}{729}+\frac{2}{2187} \right)$

2 x H = $6-\frac{2}{2187}$

2 x H = $\frac{13120}{2187}$

H = $\frac{13120}{2187}:2=\frac{6560}{2187}$

Vậy H = $\frac{6560}{2187}$