limits - $int_0^{infty} lim_{m rightarrow infty} x_m left( varepsilon right) e^{- varepsilon} mathrm{d} varepsilon$

In the expression

$$\int_0^{\infty} \lim_{m \rightarrow \infty} x_m \left( \varepsilon \right)
e^{- \varepsilon} \mathrm{d} \varepsilon$$

Is it possible to move the integral inside the Newton's iteration?

$$x_{i + 1} \left( \varepsilon \right) = x_i \left( \varepsilon \right) -
\frac{f \left( x_i \left( \varepsilon \right) \right)}{f' \left( x_i \left(
\varepsilon \right) \right)}$$

Answer

You need Dominated convergence theorem or monotone convergence theorem for exchanging limits and integrals. That is, either:

• $x_m(\epsilon)e^{-\epsilon}\le x_{m+1}(\epsilon)e^{-\epsilon} \forall \epsilon\ \forall m$ (or)



• There should exist an integrable function $f(\epsilon)$ such that $x_m(\epsilon)\le f(\epsilon)\ \forall \epsilon\ \forall m$.