# 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 $x=({x}_{1},\mathrm{\dots},{x}_{N})$ and ${a}_{\alpha}\in \u2102.$
Fixing $v\in {\mathbb{R}}^{N}$ and we can also talk of the formal power series in $t\in \mathbb{R}$

$$\begin{array}{cc}\hfill T(tv)& =\sum _{\alpha}{a}_{\alpha}{(tv)}^{\alpha}\hfill \\ & =\sum _{\alpha}{a}_{\alpha}{v}^{\alpha}{t}^{|\alpha |}\hfill \\ & =\sum _{k=0}^{\mathrm{\infty}}\left(\sum _{|\alpha |=k}{a}_{\alpha}{v}^{\alpha}\right){t}^{k}.\hfill \end{array}$$ |

###### Theorem.

Suppose $T\mathit{}\mathrm{(}x\mathrm{)}$ is a formal power series in $x\mathrm{\in}{\mathrm{R}}^{N}$. Suppose
$T\mathit{}\mathrm{(}t\mathit{}v\mathrm{)}$ is a convergent^{} power series^{} in $t\mathrm{\in}\mathrm{R}$ for
all $v\mathrm{\in}{\mathrm{R}}^{N}$. 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.

Title | formal power series converges if and only if it converges along every line |
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Canonical name | FormalPowerSeriesConvergesIfAndOnlyIfItConvergesAlongEveryLine |

Date of creation | 2013-03-22 17:42:11 |

Last modified on | 2013-03-22 17:42:11 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 5 |

Author | jirka (4157) |

Entry type | Theorem |

Classification | msc 13H05 |

Classification | msc 13B35 |

Classification | msc 13J05 |

Classification | msc 13F25 |