# What is $\mathbb Z]$? What are the double brackets?

What does $\mathbb{Z}[[t]]$ mean? Why are there double square brackets?

I can’t search through Google, because I can’t search Latex.

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That is the ring of formal power series in $t$ with integer coefficients, i.e., of $$\sum_{n=0}^\infty a_nt^n,$$ with $a_n\in\Bbb Z$, componentwise addition, and multiplication appropriately defined.

The double brackets distinguish it from $\Bbb Z[t]$, which is the ring of polynomials in $t$ with integer coefficients. We can always evaluate the members of $\Bbb Z[t]$ for any complex value of $t$, but we generally can’t evaluate members of $\Bbb Z[[t]]$ for $t\neq 0$. To my mind, the double bracket is a reminder that we need to leave the $t$ alone, and not worry about evaluation.

If $A$ is any ring, the notation $A[[T]]$ stands for the ring of formal power series with coefficients in $A$, i.e. the ring whose elements are the expressions
$$a_0+a_1T+a_2T^2+a_3T^3+\cdots$$
with the obvious sum and product.