(i) (1+\dfrac{3}{2}+\dfrac{5}{4}+\dfrac{7}{8}+\cdots)

(t_n=(2n-1)\left(\dfrac{1}{2}\right)^{n-1})

Infinite AG sum with (|r|=\frac12<1): [ S=\dfrac{a}{1-r}+\dfrac{dr}{(1-r)^2} ] Here (a=1), (d=2), (r=\frac12): [ S=\dfrac{1}{1/2}+\dfrac{2\cdot\frac12}{(1/2)^2}=2+\dfrac{1}{1/4}=2+4=6 ]

Answer: (6)

(ii) (2+\dfrac{5}{3}+\dfrac{8}{9}+\dfrac{11}{27}+\cdots)

(a=2), (d=3), (r=\frac13)

[ S=\dfrac{2}{1-\frac13}+\dfrac{3\cdot\frac13}{\left(\frac23\right)^2}=\dfrac{2}{\frac23}+\dfrac{1}{\frac49}=3+\dfrac{9}{4}=\dfrac{21}{4} ]

Answer: (\dfrac{21}{4})