$$\frac{1}{1+2}\frac{2}{1+2^{2}}+3}{1+2^{3}}+\cdot\cdot\cdot$$
$\frac{47}{15}\approx 3.133333333$
$$\frac{1}{3}\times \frac{2}{1+2^{2}}+3$$
$$\frac{1}{3}\times \frac{2}{1+4}+3$$
$$\frac{1}{3}\times \frac{2}{5}+3$$
$$\frac{1\times 2}{3\times 5}+3$$
$$\frac{2}{15}+3$$
$$\frac{2}{15}+\frac{45}{15}$$
$$\frac{2+45}{15}$$
$$\frac{47}{15}$$
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$\frac{47}{3 \cdot 5} = 3\frac{2}{15} = 3.1333333333333333$
