# Tagged Questions

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### Convergent Sequence of Analytic Functions, Do Their Derivatives Converge?

If you have a sequence of real analytic functions that converge on every compact subset in your domain, do their derivatives necessarily converge to the derivatives of the function that they converge ...
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### If $f_n(t):=f(t^n)$ converges uniformly to continuous function then $f$ constant

Let $f \colon [0,1] \to \mathbb R$ be continuous and let $f_n$ be defined by $$f_n(x):=f(x^n), \qquad x \in [0,1], \,\, n \in \mathbb N.$$ Suppose there exists a continuous function $g$ on ...
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### $\sum a_n$ converges, $a_n \in \mathbb{R}$, then there exists real sequence $b_n$ such that $b_n\rightarrow +\infty$ and $\sum a_n b_n$ converges.

Assume that $\sum a_n$ converges, $a_n \in \mathbb{R}$, then there exists real sequence $b_n$ such that $b_n\rightarrow +\infty$ and $\sum a_n b_n$ converges. Same to be easy at first thought, can ...
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### Find limit property

Let $f$ be a function on $\mathbb{R}$ satisfy: $|f(x)-f(y)|\leq|x-y|$ $\forall x,y\in\mathbb{R}$. Consider the sequence: $$u_{n+1}=\frac{u_n+f(u_n)}{2},u_0=a$$ Research the limit property of this ...
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### Prove for a monotone function, if $x_0$ is interior on interval $I$ then the one-sided limits exists

The proof goes like this: Lets suppose $f$ is nondecreasing (for nonincreasing we'll observe the function $-f(x)$). Let $x_0$ an interior point in the interval $I$, and $\left\{x_k\right\}$ an ...
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### sum of the series $\sum_{k=0}^{\infty}(k+1)(1-|x_k|)|x_n|^k$ [closed]

Let $|x_n|<1$. Is the given series convergent or which conditions do we impose on the sequence $(x_n)$ to given series be convergent? $\sum_{k=0}^{\infty}(k+1)(1-|x_k|)|x_n|^k$
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### Is $(\frac{1}{1-|x_n|})$ bounded or only bounded above if $|x_n|<1$? [closed]

Let $(x_n)$ be a sequence such that $|x_n|<1$. Is $(\frac{1}{1-|x_n|})$ bounded or only bounded above if $|x_n|<1$?
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### Is $(|x_n|^{n+1})$ convergent or bounded if $|x_n|<1$?

Let $(x_n)$ be a sequence such that $|x_n|<1$. Is $(|x_n|^{n+1})$ convergent or bounded?
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### A series with only rational terms for $\ln \ln 2$

We all know that $$\ln 2 = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}.$$ Do you know a series with only rational terms for $$\ln \ln 2 = ?$$ Let's exclude base expansions with non ...
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### sum of the series $\sum_{k=0}^{\infty}(k+1)(x_n)^k.$

Let $x_n$ be a sequence of real numbers such that $x_n\in(0,1).$ Find the sum of the series $\sum_{k=0}^{\infty}(k+1)(x_n)^k.$ My answer is $\frac{(x_n)'}{(1-x_n)^2}.$ But the term $(x_n)'$ make me ...
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### Upper hemicontinuity and closed graphs

I have a problem with the definitions of upper hemicontinuity. In particular I found a picture that makes me wonder, not only about my understanding of this concept, but of the concept and application ...
I'm reading a paper, for proving a claim it defines $$f_n(x) = \dfrac{(rx-x^2)^n}{n!}$$ when $r = \frac{a}{b}$ is a rational, and $I_n = \int^r_0 f_n(x) \cdot \sin x \cdot dx$ , and then it says ...