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Convergence of the integral $\int_0^{1/n} \frac{(1-x)^n}{1+x^n}$
The following sum is apparently infinite:
$\int_0^{1/n} \frac{(1-x)^n}{1+x^n}$
Is this correct?
A:
We have
$\frac{1-x^n}{1+x^n} = \frac{1}{1+x^n}-\frac{x^n}{1+x^n} =\frac{1}{1+x^n}-\frac{1}{1+x} = \frac{1}{(1+x)(1+x^{n-1})}$
hence
$\int_0^{1/n} \frac{(1-x)^n}{1+x^n} = \int_0^{1/n}\frac{1}{(1+x)(1+x^{n-1})}\,dx = \int_0^1\frac{1}{(1+x)(1+x^{n-1})}\,dx$
and the integral is convergent since
$\left|\frac{1}{(1+x)(1+x^{n-1})}\right|\le\frac{1}{1+|x|+|x^{n-1 0b46394aab
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