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 $j>1$, $$
N\_j=
N\_{j-1}+1\qquad D\_j=
2D\_{j-1}+1$$

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

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> What is the sum of the first $25$ term?
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Must we add them one-by-one?

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