# All Questions

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### Using Abel's test for negative coefficients

For the power series representation of $$\log(1-z)=\sum_{n=1}^{\infty}=\frac{-z^n}{n},$$ is Abel's test still valid to conclude that the $\log(1-z)$ will converge everywhere on the unit disk, ...
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### Is my alternative proof that the intersection of nested compact sets is nonempty valid?

Let $\{A_n\}_{n=1}^\infty$ be a collection of nested closed sets in a compact space $X$. Since $A_n$ is closed, it is compact, and consequently limit point compact. Let $\varepsilon > 0$ and define ...
7k views

### Given $f(x)$ its inverse function, domain and range

$f(x) = \frac{{2x + 3}}{{x - 1}},\left[ {x \in {R},x > 1} \right]$ I've got the inverse function to be: ${f^{ - 1}}(x) = \frac{{x + 3}}{{x - 2}}$ How would I go about working out the range and ...
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### If a linear system has no free variables, then it is consistent: Why is the statement false?

I recently got this question wrong on a test, and I have no clue why. How can a system be inconsistent if there are no free variables?
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### Doubt in proof of special case of implicit function theorem

I've been studying the implicit and inverse functions theorems and I've started with one special case of the implicit function theorem. The book I'm reading states the theorem as follows: ...
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### Why is the matrix norm $||A||_1$ maximum absolute column sum of the matrix.

By definition, we have: $||V||_p=\sqrt[p]{\displaystyle \sum_{i=1}^{n}|v_i|^p}$ $||A||_p=sup\frac{||Ax||_p}{||x||_p}$, $x\not=0$ and if $A$ is finite, we change sup to max. However I don't really ...
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### Zariski tangent space and $K[\epsilon]/(\epsilon^2)$ [duplicate]

I want to prove that the Zariski tangent space at $x\in X$ ($X$ is an affine scheme) is isomorphic to $Hom_K(X,K[\epsilon]/(\epsilon^2))$ (K is the residue field at $x$). I want to say that ...
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### spanning trees of an edge transitive graph

Let $G$ be an edge transitive graph. Let $t(G)$ be the number f spanning trees on $G$. Show that each edge lies in exactly $\tfrac{(n-1)t(G)}{m}$ spanning trees. Where $|V(G)|=n$ and $|E(G)=m$. ...
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### Is the Cartesian product of an infinite number of $\mathbb{Z}^+$ countable?
We know the Cartesian product of a finite number of $\mathbb{Z}^+$ is countable, for example, $\mathbb{Z}^+ \times \mathbb{Z}^+ \times \mathbb{Z}^+$, because, let $m, n, p$ be from each of the ...