# All Questions

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### I have a question about properties of series.

Let $x_n$ be a sequence of real numbers. Show that if $\sum_{n=1}^{\infty}x_n$ converges, then $\lim_{n\rightarrow\infty}\sum_{k=n}^{\infty}x_k=0.$ This is what I have done. We have by the Cauchy ...
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### Something related to maximal ideal

Let $M$ be a maximal ideal in the polynomial ring $\Bbb C[x]$. Prove that there is some $a \in \Bbb C$ such that $M$ is the ideal generated by $x-a$ This question I have no idea how to prove, hope ...
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### Denseness of normal and non-normal numbers

How to prove that normal numbers are dense ? I have read in a book that this set has full measure. so dense. Then how to argue for the no-normal numbers. They also turn out to be dense. How to argue ...
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### Showing that $|P|$ divides $|A|!$ where $P$ is a $p$-group and $A$ is maximal abelian normal

I'd appreciate verification of the following proof. This is the part following what I asked in this question. Thanks. If $A$ is as in the linked-to question above (that is, $A$ is maximal among ...
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### How to prove that the operator $(\lambda I-A)^{-1}$ exists?

Let $A:H^1(\mathbb{R})\to L^2(\mathbb{R})$ be the operator given by $Aw=w_x$, where $w_x$ denotes the weak derivative of $w$. I need help to prove that $(\lambda I-A)^{-1}$ exists and is bounded for ...
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### Ring of formal power series over a field is a principal ideal domain

If $K$ is a field, any non-zero ideal in the ring of formal power series $A=K[[X]]$ is of the form $AX^n$ with $n\geq 0$, so $A=K[[X]]$ is a principal ideal ring. I can't see why. Usually when we ...
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### Characteristic Function and Random Variable Transformation

Let $X$ be a random variable, and let $\phi_X(t)$ be its characteristic function. Let $Y = f(X)$ be a transformation of the random variable $X$ where $f$ is increasing and one-to-one. Is there a ...
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### Predicting digits in $\pi$

Is it possible to predict next digit in $\pi$ using $N$ previous digits, so on and so forth? Or is this impossible because it's irrational? Basic assumption is that the person doesn't know a ...
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### Help with vectorial analysis exercise

Let $D(0,r) := \left\{ {x \in \mathbb{R}^n: \|x\| \leq r }\right\}$ and $f:D(0,r) \rightarrow \mathbb{R}$ be a continuous differentiable function in the interior of $D(0,r)$. I'm trying to show ...
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### Fundamental lemma of calculus of variations, gradients

Let $D \subset \mathbb{R}^d$ be a smooth bounded domain. Let $C_c^\infty(D)$ denote smooth and compactly supported functions on $D$. Let $f \in [C_c^\infty(D)]^d$ be a smooth, compactly supported ...
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### Which of the following form an ideal in this ring?

Let $C(R)$ denote the ring of all continuous real-valued functions on with the operations of pointwise addition and pointwise multiplication. Which of the following form an ideal in this ring? The ...
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### convergence of a series in real analysis proof

If $\sum_{n=1}^{\infty}a_n$ converges, prove that $\sum_{n=1}^{\infty} \dfrac{n+1}{n}a_n$ converges. It's probably pretty trivial, but I have been staring at it for a while and cannot make any ...
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### If augmented matrix has zero row, does it mean infinite solutions?

Given an augmented matrix in REDUCED ROW ECHELON FORM representing a general system of linear equations (does not have to homogeneous), if the augmented matrix has a zero row, does it imply that the ...
### If $(x - 2)$ is a factor of $x^3 + ax^2 -6x -4$, then find $a$.
If $(x - 2)$ is a factor of $x^3 + ax^2 - 6x - 4$, then find $a$. This is regarding polynomials. The answer is $a = 2$. Could someone please provide the working out and help me out on this please. ...