# 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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### Use Taylor Theorem and Taylor Expansion to Prove

Let $g: R\rightarrow R$ be a twice differentiable function satisfying $g(0)=1, g'(0)=0$ and $g''(x)-g(x)=0$, for all $x$ in R Fix $x$ in R. Show that there exists $M>0$ such that for all natural ...
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### Is $f$ integrable in $L(X,\mathcal{X},\mu)$

Is $f$ integrable $L(X,\mathcal{X},\mu)$ $\mu(E)=\sum_{n\in E\cap\mathbb{N}} |n^2+n-6|$ $f:\mathbb{R}\rightarrow \mathbb{R_+}\cup\infty$ $f=(x-2)^{-4}$
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### To express a integrable function as difference.

Assume $f$ is an integrable function on $[0,1]$. I want to find functions $g$ and $h$, so that $f=g-h$ almost everywhere. The functions $g$ and $h$ should be pointwise limits of continuous functions ...
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### Nice convergent subsequence of $\cos(n)$.

This question is related to a few questions which have been posted on the website : Is there a limit of $\cos(n!)$ Converging subsequence on a circle The limit of $\sin(n!)$ Because of the ...
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### Stirling approximation / Gamma function

Is it possible to obtain the Stirling approximation for the factorial by using the gamma function ? $$\Gamma(z) = \lim_{n \to +\infty} \frac{n! n^{z}}{z(z+1) \dots (z+n)}$$ Any hint ?
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### Unordered summation

I have a question regard unordered sumation. Definitions: Let $X$ be a set (possible uncountable) and let $f: X\rightarrow \mathbb{R}$ be a function. We say that $\sum _{x\in X}f(x)$ converges ...
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### If $f$ is continuous on $[0,1]$ and if $\int\limits_{0}^{1} f(x) x^n dx = 0, (n=0,1,2,…)$, prove that $f(x)=0$ on $[0,1]$.
If $f$ is continuous on $[0,1]$ and if $\int\limits_{0}^{1} f(x) x^n dx = 0, (n=0,1,2,...)$, prove that $f(x)=0$ on $[0,1]$. This is what I have, how does it look? Proof: Let $P(x)$ be any ...