(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)