Finding $\lim_{N\to\infty}\int_{0}^{\infty}1-\left(1-\exp\left(-x^2\right)\right)^{N}\text{d}x$

Question

Is my answer correct?

Since $\forall N\in\mathbb{N}^{*}$, $x \geq 0$, $\left| 1-\left(1-\exp\left(-x^2\right)\right)^{N} \right|<1$, so by the Dominated Convergence Theorem, we have

$\lim_{N\to\infty}\int_{0}^{\infty}1-\left(1-\exp\left(-x^2\right)\right)^{N}\text{d}x=\infty$

Also, I have calculated the integral by integrating over the binomial expansion as

$\int_{0}^{\infty}1-\left(1-\exp\left(-x^2\right)\right)^{N}\text{d}x= N\sqrt{\frac{\pi}{8}}\sum_{k=0}^{N-1}\binom{N-1}{k}\frac{\left(-1\right)^k}{\left(k+1\right)^{\frac{3}{2}}},$

but it seems more involved to find the limit under this new expression.

Any suggestions are very well appreciated.

Answer 1

1 isn’t integrable over the entire real line so you can’t apply DCT. You can however apply the Monotone Convergence Theorem to get the same result.