Tagged Questions

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Power series centered at $x =0$

I have this question in my advanced calculus textbook. Give an example of a power series centered at $x = 0$ that converges on $[-2,2]$ but not absolutely on the entire interval $[-2,2]$, and ...
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Finding a function without knowing its structure but some conditions

I'm trying to find a function who meets this conditions but have no idea where to start. Just think it may be related to the function $Ca^{-\left(x-\mu\right)^2}$, If it really has this structure (or ...
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Extreme Value Theorem over the Real Numbers

I'm stuck on where to start with this. I can tell it is to do with the extreme value theorem, but past that point I'm stuck. Any help would be appreciated. If $f(x)$ is continuous on $\mathbb{R}$, ...
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Is there an analytic function satisfying $\,\,f\big(\!\frac 1 n\!\big)=\frac 1 {\sqrt{n}},\,$ for every $\,n\in\mathbb N$?

Is there a function that is analytic in an open neighbourhood of $z=0$ and satisfies $$f\left(\!\dfrac 1 n\!\right)=\dfrac 1 {\sqrt{n}},$$ for all natural number $n$? I guess this problem requires ...
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To show a function is differentiable

Let $f, g : \mathbb{R}^n → \mathbb{R}^m$ . Suppose that $f$ is differentiable at $p ∈ R$ , that $f (p) = 0$ and that $g$ is continuous at $p$ . Let $h(x) = f (x) g(x)$ (again we use the standard ...
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A rearrangement of an absolutely convergent complex series is also absolutely convergent

I just completed the following proof. Is it valid? Let $\sum_{k=1}^{\infty} a_k$ be an arbitrary convergent series that also converges absolutely. Then $\sum_{k=1}^{\infty} a_k \in \mathbb{C}$ and ...
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the best constant in an inequality?

I learnt how to show the below inequality by C-S inequality: k is from $0$ to $\infty$ If $\sum a_{k}^{2}9^{k}\le 5$ then $\sum |a_{k}|2^{k}\le 3$. next,I tried to show that 3 is the best possible ...
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Decimal representation

I need to prove that, given ${a,n}$ integers with $a<10^n$, $\frac a {10^n}$ has two different decimal representations. I know that this is related to the fact that $0,99999... =1$, and I know how ...
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Find critical points of sin(x*y)

so I got this homework problem that I was having trouble with. The problem is: Let $f(x, y) = \sin(xy)$ defined on all of $\mathbb{R}^2$. Find the critical points of $f$ and classify them as local ...
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Proving differentiability

I'm trying to do Spivak's Calculus on Manifold excersise 2-4. It goes as follows: Let $g$ be a continuous real valued function on the unit circle $\{x\in\mathbb{R}^2:||x||=1\}$ such that ...
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Convergence of an infinite sum

Is it possible to use the comparison test for convergence in the following series? $$\sum_{n=1}^\infty \sin \frac 1 n$$ The exercise says that I should find a linear function $f(x)$ that satisfies ...
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How to find $\alpha_{k} \uparrow \infty$ such that $\sum |\alpha_{k}a_{k}|< \infty$.

let $\sum |a_{k}|< \infty$. How to show that there exists a sequence $\alpha_{k} \uparrow \infty$ such that $\sum |\alpha_{k}a_{k}|< \infty$. Could you please help me with this question.
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An application of Cauchy-Schwarz ineq. on infinite series

If $\sum a_{k}^{2}9^{k}\le 5$ then $\sum |a_{k}|2^{k}\le 3$. sums are from $0$ to $\infty$. could you please help with this question.
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How can I check the convergence of the sequence? Does it diverge?

How can I check the convergence of the sequence $\frac{1}{\sqrt{n^2+1}}+\frac{2}{\sqrt{n^2+2}}+\cdots+\frac{n}{\sqrt{n^2+n}}$? I think that it diverges,because it is bounded below from ...
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Finding upper and lower darboux sums

Let $f : [0,1] \to \mathbb{R}$ be defined by $f(x) = 1$ if x is rational, and $f(x) = 0$ is x is irrational. Let $P$ be a partition of $[0,1]$. Find the lower and upper Darboux sums, $L(f,P)$, and ...
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How to choose $f\in C^{2}(\mathbb{R})$ with compact support and takes value 1 on connected compact set?

Let $0< \delta < \pi$. My questions: (1) How to construct(choose/method) $f\in C^{2}(\mathbb R)$(= First two derivatives ($f' \ \text{and} \ f''$) of $f$ on $\mathbb R$ exists and are ...
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Limits of general functions

If $$\lim_ {x\to 0} \ f(x) = 0$$ Then prove (i) $$\lim_{x \to 0} \ f(x^2) = 0$$ and prove (ii) $$\lim_{x \to 0} \ f(2x) = 0$$ I'm aware I need to use the epsilon-delta Cauchy definition, so I have ...
Determine for what value of $x$ the series converges $∑_{n=1}^∞ \frac{(3^n+(-2)^n )}{n} (x+1)^n$ Here is what I got Using the ratiotest, I got $D_n =\frac{\frac{(3^{n+1}+(-2)^{n+1} ... 1answer 31 views convergence of series, double factorial against power function? It is known that the series $$\sum_{n=1}^{\infty} \frac{C^n}{n!}<\infty$$ for any$C>0$, that is, the factorial kills the Power function. I wonder now if $$\sum_{n=1}^{\infty} ... 0answers 31 views Computing an (iterated) integral I am having problems to find a closed form depending on n of the following integral.$$ \int_{t_{0}\ <\ w_{1}\ <\ \cdots\ <\ w_{n}\ <\ t}\ {{\rm d}w_{1}\cdots {\rm d}w_{n} \over ... 1answer 34 views $T_n \rightarrow T$then we have$||T|| \le liminf(||T_n||)$I know how to show that a cauchy sequence of linear continuous operators$T_n:X \rightarrow Y$has a limit that is also such an operator(if Y is a Banach space), but I found this relation here too ... 2answers 52 views Making$g(x)=x^2\operatorname{sgn}(x)$continuous at$0$. How would the function$g(x) = x^2 \operatorname{sgn} (x) $be defined at$x= 0 $so that it is continuous there? This makes no sense to me. It isn't, period. Why would it be? I can't re-define the ... 1answer 83 views real analysis part 2 I need help I'm so clueless on how to do this one Suppose$\sum n a_n$is absolutely convergent and define$f(x) = \sum_{n=1}^\infty a_n \cos{n x}$define$g(x) = \sum_{n=1}^\infty a_n ...
Given any sequence whatsoever of real numbers (a_r), there is a smooth function $f: \mathbb{R} \to \mathbb{R}$ such that $f^{(r)}(0) = a_r$. Pugh's hint says to try $f=\sum \beta_k(x)a_kx^k/k!$, ...