Intereting Posts

Finding $\int_{0}^{\pi/2} \frac{\tan x}{1+m^2\tan^2{x}} \mathrm{d}x$
Sums of binomial coefficients
Odd Binomial Coefficients?
How come such different methods result in the same number, $e$?
Representation of the elements of a normed vector space that has a dense subset.
Proving uniqueness
Inverse of $x\log(x)$ for $x>1$
The number $2^{29}$ has exactly $9$ distinct digits. Which digit is missing?
Necessary and Sufficient Conditions for Random Variables
Find location and width of boundary layer
An arctan integral $\int_0^{\infty } \frac{\arctan(x)}{x \left(x^2+1\right)^5} \, dx$
$M,N\in \Bbb R ^{n\times n}$, show that $e^{(M+N)} = e^{M}e^N$ given $MN=NM$
Approaching the circumference of a circle
How prove this $|A||M|=A_{11}A_{nn}-A_{1n}A_{n1}$
Convergence of infinite product $\prod_{n=2}^\infty (1- \frac 1n) $

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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- Show that if $A^{n}=I$ then $A$ is diagonalizable.
- Show that a matrix $A$ is singular if and only if $0$ is an eigenvalue.
- system of matrix equations
- If $B$ is a maximal linearly independent set in $V$ then $B$ is a basis for $V$

- Let $f: V_n \to V_n$ be an endomorphism, prove $\text{dim}(\text{Ker}f \cap \text{Im} f) = r(f) - r(f^2)$
- Eigenvalues of a rectangular matrix
- Proof If $AB-I$ Invertible then $BA-I$ invertible.
- Questions about rank and eigenvalues of a matrix
- Importance of rank of a matrix
- Importance of Linear Algebra
- Differences between infinite-dimensional and finite-dimensional vector spaces
- How to prove that if $\det(A)=0$ then $\det(\operatorname{adj}(A))=0$?
- Proving that for each two parabolas, there exists a transformation taking one to the other
- Differentiation with respect to a matrix (residual sum of squares)?

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.

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