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Determine if the following series is convergent or divergent $\sum_{n=1}^{\infty}\frac{ \sin{nx}}{n}$
6 votes

You can use Dirichlet criteria because $\sum_{k=1}^{n}\ \sin{kx}$ is bounded. To see this, notice that $\sum_{k=1}^{n}\ \sin{kx}$ = $Im(\sum_{k=1}^{n}\ e^{ikx})$.

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How to take the exponential of a non-diagonal matrix
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5 votes

Write $A=2I+N$ where $$N=\begin{bmatrix} 0 & 1 \\ 0 & 0 \\ \end{bmatrix}$$ Then $\exp(A)=\exp(2I)\exp(N)$ since $I$ and $N$ commute. $\exp(N)$ is not too difficult to compute as $N^2=0$...

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Find values of $f'(2)$ for given expression
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5 votes

Hint Write $f(x)=(x-2)g(x)$ Then $f'(x)=g(x)+g'(x)(x-2)$.

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Matrix with irreducible minimal polynomial gives rise to a field
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4 votes

The two not obvious points needed to prove are : $a,b \in E \implies ab \in E$ Every $a \in E$ such that $a \neq 0$ is invertible. Let's start with a useful lemma: Lemma: Let $P \in K[X]$. Then $P(...

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If $f(x) = \frac{\cos x + 5\cos 3x + \cos 5x}{\cos 6x + 6\cos4x + 15\cos2x + 10}$then..
4 votes

$f(x) = \frac{u(x)}{v(x)}$ so $f(x) v(x) = u(x)$ Differentiating gives: $f'(x) v(x) + v'(x) f(x) = u'(x)$ Differentiating again: $f''(x) v(x) + f'(x) v'(x) + v''(x)f(x)+f'(x)v'(x) = u''(x)$ ...

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Proof that $Cov(aX+b, Y+Z)=aCov(X,Y)+aCov(X,Z)$
4 votes

It is much faster to use the fact that $Cov$ is a bi-linear map. Hence $Cov(aX+b,Y+Z) = a Cov(X,Y+Z) + Cov(b,Y+Z)$ (linearity w.r.t first variable) $b$ is constant so $Cov(b,Y+Z)=0$. Then $Cov(aX+b,...

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Fourier Series of a sum of two functions
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4 votes

Yes because of the linearity of integration.

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convergence of $a_n$ knowing that $a_n^n$ converges
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4 votes

Take $(a_n)$ defined by : $a_n = 0$ if n is even. $a_n = \frac{1}{2}$ if n is odd. This sequence satisfies (3) but it does not converge.

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Limits at infinity of a function with convergent improper integral
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3 votes

For (2) by the same reasoning by contradiction, there exist $L>0$ and $x_0$ such that for all $x>x_0$, $|f(x)| \geq L$. Because $f$ is continuous, either $f(x) \geq L$ for all $x>x_0$ or $f(...

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How to find $E[e^{\lambda X}]$ for a random variable $X$?
3 votes

If is $f$ is the pdf of $X$ then $\mathbb{E}[e^{\lambda X}] = \int e^{\lambda x}f(x) dx$.

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Showing that this is a martingale.(4.13 in Øksendals SDE)
3 votes

Let's show that $\mathbb{E}[\int_0^t (e^{Z_s}(1-X_su_s))^{2}ds] < \infty $ By Cauchy-Schwarz inequality: $\mathbb{E}[\int_0^t (e^{Z_s}(1-X_su_s))^{2}ds] \leq \mathbb{E}[\int_0^t e^{4 Z_s}ds]^{\frac{...

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Suppose $\int_0^9f(t)dt=12$. Then is it true that $\int_0^3f(3x)dx = 12$
3 votes

Hint: Try the change of variable $t=3x.$

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$\det(I+\epsilon V)=1+\operatorname{trace}(V)\epsilon+O(\epsilon^2)$
3 votes

Hint : You can see this easily by triangulating $V$. (which is always possible in $M_n(\mathbb{C})$) You would get that $\det(I+\epsilon V) = \textstyle \prod_{i=1}^{n} (1+\epsilon \lambda_i)$ where $...

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Proving a set of vectors is independent
2 votes

From $$ \alpha{\begin{pmatrix} 1\\ 1\\ 0\\ \end{pmatrix}} +\beta{\begin{pmatrix} 1\\ 0\\ 1\\ \end{pmatrix}}= {\begin{pmatrix} 0\\ 0\\ 0\\ \end{pmatrix}}$$ If you consider the third ...

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When arg max - f(x) = arg min f(x)
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2 votes

Hint $f(x) \leq f(y) \Leftrightarrow -f(x)\geq -f(y)$

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Integration problem. Does $\left|\int_0^\infty f(x)dx\right|=0$ imply $f=0$ over $(0,\infty)$?
2 votes

Take $f(x)=\sin(x)$ if $x\in[0,2\pi]$ and $f(x)=0$ otherwise. Then $f$ is continuous and $|\int_0^\infty f(x)dx|=0$ but $f \neq 0$.

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Prove that the derivative of $x^w$ is $w x^{w-1}$ for real $w$
2 votes

Hint Use induction along with the product rule.

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How to use the completeness here
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2 votes

Let $(h_n)$ a sequence in $V$ converging in $\mathbb{C}$. Let $u_n = \frac{f(z_0+h_n)-f(z_0)}{h_n}$ The bounded $\sup$ condition implies $||u_p - u_q|| \leq M |h_p-h_q|$ Hence $(u_n)$ is a Cauchy ...

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Find $I$ in $\frac{\overline{SIX}}{\overline{NINE}}=\frac23$
2 votes

Because $N=1$, the equation is: $300S + 3X = 2020 + 170I +2E$ Let's write it $R(S,X) = L(I,E)$. We notice that $L(5,0)=2870$ and $R(9,9)=2727 < L(5,0)$ so $I<5$. Let's check the remaining 5 ...

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Checking linear dependence of two vectors in $\mathbb{R}^4$?
2 votes

I think the fastest way would be to do it by contradiction. Let $X=(3,-1,1,1)$ and $Y=(1,3,-1,1)$ Assume $X$ and $Y$ are linearly dependent. Then there is $a$ such that $X= aY$. Using this on the ...

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Determine whether $f''(x)>0$ or not in a given interval
2 votes

Hint: Let $g$ =$f'$. You have shown that there exists $x_1$ and $x_2$ such that $g(x_1) = g(x_2)$. Apply Rolle's theorem.

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Finding other eigenvalue of matrix given one eigenvalue
2 votes

1) Let $\lambda_1, \lambda_2$ the eigenvalues of $A$. We know that $\lambda_1=2$. We have that $\det A = \lambda_1 \lambda_2 $ By assumption, $\det A = 1$ so $1= 2\lambda_2$ and hence $\lambda_2 = ...

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Vector space $V\cong \mathbb R^n$ or $\cong \mathbb C^n$.
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2 votes

Yes. More generally, if $E$ and $F$ are both $\mathbb{K}$-vector spaces of dimension $n$ then $E \simeq F$.

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Prove this for any $k>0$
2 votes

Work by induction. For $k=1$ it is true as $e >1$ Assume it is true for $k$. Then $(k+1)! = (k+1) k! > k k!> k \frac{k^k}{e^k} = e \frac{k^{k+1}}{e^{k+1}} > \frac{k^{k+1}}{e^{k+1}}$ ...

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Diagonalisation proof
2 votes

Hint : $A^{n}=(P^{-1} D P)^{n}=P^{-1} D^{n} P$

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Does $\sum_{n=1}^\infty \left(\frac{1}{n} - 1\right)^n$ and $\sum_{n=1}^\infty \left(\frac{1}{n} - 1\right)^{n^2}$ converge?
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2 votes

Let $a_n = ( \frac{1}{n} - 1)^{n^2}$ Then $|a_n| = (1- \frac{1}{n})^{n^2} =\exp(n^2 \ln (1-\frac{1}{n})) \sim e^{-n+\frac{1}{2}}$ $ \sum e^{-n+\frac{1}{2}}$ converges so $ \sum |a_n|$ converges ...

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What can the operator $T$ be?
Accepted answer
2 votes

$T^3 = T$ implies that T is diagonalizable. Hence T is diagonalizable and 0 is its only eigenvalue. What can you conclude ?

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If X,Y are both bounded and closed, does this imply that X+Y is bounded and closed?
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2 votes

You are right that it is straightforward that $X+Y$ is bounded. In fact, $X+Y$ is also closed: Let $(u_n)$ a sequence of $X+Y$ elements such that $\lim(u_n) = u$. We want to show that $u \in X+Y$. ...

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$(\mathcal L(X,Y),\|\cdot \|)$ is complete if $Y$ is complete.
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2 votes

The last implication directly stems from : $\forall n\geq N : \forall x\in X, x\neq0,\frac{\|(\hat T-T_n)x\|}{\|x\|}\leq \varepsilon$ Taking the supremum for all $x\neq0$, we get by definition : $\|\...

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Prove that $\left(\sum^n_{k=1}x_k\right)\left(\sum^n_{k=1}y_k\right)\geq n^2$
1 votes

You can use Cauchy-Schwarz inequality with $u_i = \sqrt{x_i}$ and $v_i=\frac{1}{\sqrt{x_i}}$.

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