We study invariance principles and convergence to a Gaussian limit for stochastic series of the form $S(c,Z)=\sum_{m=1}^{\infty }\sum_{α_{1}<...<α_{m}}c(α_{1},...,α_{m})\prod_{i=1}^{m}Z_{α_{i}}$ where $Z_{k}$, $k\in \mathbb{N}$, is a sequence of centred independent random variables of unit variance. In the case when the $Z_{k}$'s are Gaussian, $S(c,Z)$ is an element of the Wiener chaos and convergence to a Gaussian limit (so the corresponding nonlinear CLT) has been intensively studied by Nualart, Peccati, Nourdin and several other authors. The invariance principle consists in taking $Z_{k}$ with a general law. It has also been considered in the literature, starting from the seminal papers of Jong, and a variety of applications including $U$-statistics are of interest. Our main contribution is to study the convergence in total variation distance and to give estimates of the error.