设函数 $f(x)=\int_0^x \frac{\ln (1+t)}{1+e^{-t} \sin ^3 t} \mathrm{~d} t,(x>0)$, 证明级数 $\sum_{n=1}^{\infty} f\left(\frac{1}{n}\right)$ 收敛, 且 $\frac{1}{3} < \sum_{n=1}^{\infty} f\left(\frac{1}{n}\right) < \frac{5}{6}$.
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