# All Questions

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### What function does the power series $\sum_{n=0}^{\infty}\frac{z^n}{n^2}$ converge to in its disc of convergence?

For the power series $\sum_{n=0}^{\infty}\frac{z^n}{n^2}$, its radius of convergence is 1 which implies that this series is absolutely convergent in the the unit ball $\{z:|z|<1\}$. Since it is ...
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### determining the limits in $u$-substitution

I have a fundamental question regarding $u$-substitution: Suppose that want to calculate the integral $$\int_{a}^{b}f(x)\,dx$$ and we wish to use a $u$-substitution of the form $x=g(u)$ (rather ...
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### Determinant of determinant is determinant?

Looking at this question, I am thinking to consider the map $R\to M_n(R)$ where $R$ is a ring, sending $r\in R$ to $rI_n\in M_n(R).$ Then this induces a map. $$f:M_n(R)\rightarrow M_n(M_n(R))$$ Then ...
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### If a subset S of a vector space, V spans V then there exists a subset of S that also spans V. Prove?

Additional related question: Can span of a subset, S of a vector space, V ever be a superset of V. Answer is No! Because V will no longer be a vector space then as V will not be closed under vector ...
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### Find $\int_{C}{\bf{F}}\cdot d{\bf{s}}$ through the line segment

Let $F=\left[\frac{x}{x^2+y^2},\frac{y}{x^2+y^2}\right].$ Let $C$ by the curve consisting of the line segments from $$(-1,0)\to (0,-2)\to (2,0)\to (3,4)\to (0,5)\to (-1,0)$$ Find ...
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### Factoring $12e^{2x} - 32e^x + 16$

Can you help me solve the quadratic equation $12e^{2x}-32e^x+16$ by factoring please?
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### Ionofs Problem Solving

This was a practice question given to me but I can't seem to find the answer please help! Mostly need help with b,c,d. The Ionof (Integer on number of factors) of an integer is the integer divided by ...
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### State the domain of $f^{-1}$

$$f(x)=\sqrt{2x+5}$$ $$x \geq -2.5$$ State the domain of $f^{-1}$ \begin{align} \ x & = \sqrt{2y+5} \\ \ \Rightarrow f^{-1}(x) & = \frac{x^2-5}{2} \\ \end{align} The Mark Scheme says that ...
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### Determining the degree of a root of unity over a cyclotomic expansion

For $\xi_{n} = e^{2\pi*i/n}$ , determine the degree of $\xi_{7}$ over the field $\Bbb{Q}(\xi_{3})$ How would I approach this problem? I'm having trouble starting this problem and can't find any ...
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### Prove that the set of matrices $\{I,A,A^2,\ldots A^m\}$ is linearly independent.

Let $A$ denote the adjacency matrix of a connected graph $G$ with $n$ vertices and $e$ edges.If $i$ and $j$ are vertices of $G$ with $d(i,j)=m$. Then prove that the set of matrices ...
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### Dense Subgroup Of Reals with addition

let $G$ be a non-trivial subgroup of $(R,+)$. Prove that $G$ to be a dense or $(nZ)$. I know this result is true of course .But problem is to prove it. Please hint to prove it.
I have a question concerning the the inverse mapping in the image . text extracted from Casella Statistical inference $g^{-1}(A) = \{ x \in \chi : g(x) \in A\}$ I know the idea that they want to ...
### Why is the function $f_n(x)=\sum_{k=0}^n\frac{(-1)^k}{2^k}\lfloor 2^kx\rfloor$ bounded for some $n$ but unbounded for others?
The function is unbounded for $n=0,1,2,4$ but bounded for $n=3,5,6,7,8$, at least if desmos.com is to be believed. https://www.desmos.com/calculator/lyre5002v0 It's past my bedtime so I have not ...