For a holomorphic function f in the open unit disc $\mathbb{D}$ and $\zeta \in \mathbb{D}$, $S_n(f,\zeta )$ denotes the n-th partial sum of the Taylor development of f at ζ. Given an increasing sequence of positive integers $\mu =({\mu }_n)$, we consider the classes $\mathcal{U}(\mathbb{D},\zeta )$ and ${\mathcal{U}}^{(\mu )}(\mathbb{D},\zeta )$ of holomorphic functions f in $\mathbb{D}$ such that subsequences of the partial sums $\{S_n(f,\zeta ):n=1,2,\dots \}$ and $\{S_{{\mu }_n}(f,\zeta ):n=1,2,\dots \}$ respectively approximate all polynomials uniformly on the compact sets $K\subset \{z\in \mathbb{C}:|z|\ge 1\}$ with connected complement. We show that these two classes of universal Taylor series coincide if and only if $\underset{n\to +\infty }{\mathrm{lim\hspace{0.17em}sup}}\left(\frac{{\mu }_{n+1}}{{\mu }_n}\right)<+\infty$. In the same spirit, we prove that, for $\zeta \ne 0$, the equality ${\mathcal{U}}^{(\mu )}(\mathbb{D},\zeta )={\mathcal{U}}^{(\mu )}(\mathbb{D},0)$ holds if and only if $\underset{n\to +\infty }{\mathrm{lim\hspace{0.17em}sup}}\left(\frac{{\mu }_{n+1}}{{\mu }_n}\right)<+\infty$. Finally we deal with the case of real universal Taylor series.

对于开放单元光盘中的全纯函数f$\mathbb{d}$ 和 $\zeta \in \mathbb{d}$， ${\mathrm{小号}}_{ñ}（F，\zeta ）$表示f在ζ处的泰勒展开的第n个子和。给定正整数的递增序列$\mu =（{\mu }_{ñ}）$，我们考虑课程 $\mathcal{ü}（\mathbb{d}，\zeta ）$ 和 ${\mathcal{ü}}^{（\mu ）}（\mathbb{d}，\zeta ）$全纯函数˚F在$\mathbb{d}$ 这样部分和的子序列 $\{{\mathrm{小号}}_{ñ}（F，\zeta ）：ñ=\mathrm{1个}，2，\dots \}$ 和 $\{{\mathrm{小号}}_{{\mu }_{ñ}}（F，\zeta ）：ñ=\mathrm{1个}，2，\dots \}$ 分别对紧集上的所有多项式求均值 $ķ\subset \{ž\in \mathbb{C}：|ž|\ge \mathrm{1个}\}$与连接的补码。我们证明，只有当且仅当这两种通用泰勒级数重合$\underset{ñ\to +\infty }{\mathrm{lim\; sup}}（\frac{{\mu }_{ñ+\mathrm{1个}}}{{\mu }_{ñ}}）<+\infty$。本着同样的精神，我们证明$\zeta \ne 0$，平等 ${\mathcal{ü}}^{（\mu ）}（\mathbb{d}，\zeta ）={\mathcal{ü}}^{（\mu ）}（\mathbb{d}，0）$ 仅当且仅当成立 $\underset{ñ\to +\infty }{\mathrm{lim\; sup}}（\frac{{\mu }_{ñ+\mathrm{1个}}}{{\mu }_{ñ}}）<+\infty$。最后，我们处理真正的通用泰勒级数的情况。