It is perfectly fine to do the following:(adsbygoogle = window.adsbygoogle || []).push({});

##\displaystyle{\int_{-\infty}^{\infty}\ d\phi\ e^{-\phi^{2}/2}e^{-\lambda \phi^{4}/4!} = \int_{-\infty}^{\infty}e^{-\phi^{2}/2}\sum\limits_{n=0}^{\infty}}\frac{(-\lambda\phi^{4})^{n}}{(4!)^{n}\ n!}=\sum\limits_{n=0}^{\infty}\frac{(-1)^{n}\lambda^{n}}{(4!)^{n}n!}\int_{-\infty}^{\infty}e^{-\phi^{2}/2}\phi^{4n}##

and then to continue with the integration, but the following is not valid:

##\displaystyle{\int_{-\infty}^{\infty}\ d\phi\ e^{-\phi^{2}/2}\ \phi^{4n}=\int^{\infty}_{-\infty}d\phi\ \phi^{4n}\ \sum\limits_{m=0}^{\infty} \frac{(-1)^{m}\ \phi^{2m}}{2^{m}\ m!} = \sum\limits_{m=0}^{\infty} \frac{(-1)^{m}}{2^{m}\ m!} \int^{\infty}_{-\infty}d\phi\ \phi^{4n+2m}}##

The reason is that the integral after the Taylor expansion is not convergent, but it would be helpful if you could provide details.

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# A Convergence properties of integrals

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