<s>[INST] Limit of definite integral sequence This is taken from an admission exam textbook at our local construction university :
$$ \lim_{n\to\infty} n\int_{1}^{2} \frac{1}{x^2(1+x^n)}dx = ?$$
I tried finding tight bounds in order to use the sandwich theorem, but that got me nowhere. I also tried using Lebesgue's Dominated Convergence Theorem, but I ended up with nothing once more.
Do you have any ideas?
 [/INST] Using 
\begin{align}
\frac{1}{1+x^{n}} = \frac{1}{x^{n}} \, \frac{1}{1 + \frac{1}{x^{n}}} = \frac{1}{x^{n}} \, \sum_{k=0}^{\infty} \frac{(-1)^{k}}{x^{kn}}
\end{align}
then
\begin{align}
I_{n} &= \int_{1}^{2} \frac{dx}{x^{2} \, (1+x^{n})} \\
&= \int_{1}^{2} \frac{1}{x^{n+2}} \, \left( 1 - \frac{1}{x^{2n+2}} + \cdots \right) \, dx \\
&= - \frac{1}{n+1} \left[ \frac{1}{x^{n+1}} \right]_{1}^{2} + \frac{1}{2n+1} \, \left[ \frac{1}{x^{2n+1}} \right]_{1}^{2} + \cdots \\
&= \frac{1}{n+1} \, \left(1  - \frac{1}{2^{n+1}} \right) + \frac{1}{2n+1} \, \left( \frac{1}{2^{2n+1}} - 1 \right) + \cdots \\
&= \sum_{k=0}^{\infty} \frac{(-1)^{k}}{nk+n+1} \, \left( 1 - \frac{1}{2^{nk+n+1}} \right)
\end{align}
Now,
\begin{align}
\lim_{n\to\infty} \left\{ n \, I_{n} \right\} = \sum_{k=0}^{\infty} \frac{(-1)^{k}}{k+1} = \ln 2. 
\end{align}
</s>