The series $\frac{1}{2}+\frac{2}{5}+\frac{3}{11}+\frac{4}{23}+...$ Consider the expression $\frac{1}{2}+\frac{2}{5}+\frac{3}{11}+\frac{4}{23}+...$
Denote the numerator and the denominator of the $j^\text{th}$ term by $N_{j}$ and $D_{j}$, respectively. Then, $N_1=1$, $D_1=2$, and, for every $j>1$, $$
N_j=
N_{j-1}+1\qquad D_j=
2D_{j-1}+1$$

What is the $50^\text{th}$ term?

Must we evaluate that term-by-term until we reach the $50^\text{th}$ term?

What is the sum of the first $25$ term?

Must we add them one-by-one?

What is the exact value of the sum to $\infty$?


If $d_{j+1}=2d_j+1$ then $d_{j+1}+1=2(d_j+1)\Rightarrow d_n+1=(d_1+1)2^{n-1}=3\cdot2^{n-1}$.
So 50th term is $\frac{50}{3\cdot2^{49}-1}$
