formal power series converges if and only if it converges along every line

Suppose $T(x)$ denotes the formal power series $\\sum_{\\alpha}a_{\\alpha}x^{\\alpha},$ using the multi-index notation,where $x=(x_{1},\\ldots,x_{N})$ and $a_{\\alpha}\\in\\mathbb{C}.$ Fixing $v\\in{\\mathbb{R}}^{N}$ and we can also talk of the formal power series in $t\\in\\mathbb{R}$

\\begin{split}\\displaystyle T(tv)&\\displaystyle=\\sum_{\\alpha}a_{\\alpha}(tv)^{%\\alpha}\\\\&\\displaystyle=\\sum_{\\alpha}a_{\\alpha}v^{\\alpha}t^{\\lvert\\alpha\\rvert}\\\\&\\displaystyle=\\sum_{k=0}^{\\infty}\\left(\\sum_{\\lvert\\alpha\\rvert=k}a_{\\alpha}v%^{\\alpha}\\right)t^{k}.\\end{split}

Theorem.

Suppose $T(x)$ is a formal power series in $x\\in{\\mathbb{R}}^{N}$ . Suppose $T(tv)$ is a convergent power series in $t\\in\\mathbb{R}$ forall $v\\in{\\mathbb{R}}^{N}$ . Then $T$ is convergent.

The other direction, if $T(x)$ converges then $t\\mapsto T(tv)$ converges, is obvious.

References

1 M. Salah Baouendi,Peter Ebenfelt,Linda Preiss Rothschild.,Princeton University Press,Princeton, New Jersey, 1999.