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This question already has an answer here:

I know that $\mathbb{C}[x]$ stands for polynomial in $x$ with coefficients from $\mathbb{C}$, so that is

$$a_0+a_1x+a_2x^2+ \dots a_nx^n \in \mathbb{C}[x]$$

Than what is $\mathbb{C}[[x]]$?

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marked as duplicate by Cameron Buie, Eric Wofsey, Joel Reyes Noche, user223391, user91500 Sep 15 '15 at 8:11

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    $\begingroup$ $\mathbb C[[x]]$ is usually the notation for the ring of formal power series: $a_0+a_1x+a_2x^2+\dots$. In particular, it contains $\mathbb C[x]$. $\endgroup$ – Thomas Andrews Sep 14 '15 at 12:10
  • $\begingroup$ $[[x]]$ is used to signify that you aren't limited to a finite number of terms, i.e. you're allowed "infinite" degree polynomials (without any convergence conditions) (these are called formal power series in the variable $x$ over $\Bbb C$). $\endgroup$ – Arthur Sep 14 '15 at 12:10
  • $\begingroup$ And a big difference is that while the units of $\Bbb C[x]$ are just the nonzero constants, the units of the power-series ring are a much larger group. $\endgroup$ – Lubin Sep 14 '15 at 13:13
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It is the ring of formal power series with coefficients in $\mathbb{C}$.

See: https://en.wikipedia.org/wiki/Formal_power_series

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