# Tagged Questions

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### Pointwise convergence and uniform

There is a norm $\Vert .\Vert$ in the space $C([a,b],\mathbb{R})$ such that convergence pointwise implies convergence in norm $\Vert .\Vert$ ? I think not because if there would be the natural ...
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### Prove this monotone sequence has a bound, thus it converges.

Let $r>0$ and $\frac{r^n}{n!}$ Prove that it converges. I know that it is eventually decreasing, so it is monotone. How do I get a bound for it to show that it converges? Also how would I go ...
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### Prove Uniform Convergence of Series of functions-Help?

Let F0 be a bounded Riemann integrable function on [0, 1]. For n ā N, define $F_n(x)$ on [0,1] by $F_n(x)$ = $\int_{0}^{x}$ $F_{n-1}(t)$ dt 1) Prove that for all nā N and xā [0,1], we have ...
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### Absolute and conditional convergence of a series with $\sin(x)$

I have to explore absolute and conditional convergence of this function series I tried to find $a(n)$ and $a(n+1)$ terms of the series and then divide it and take a limit. But I've got nothing. ...
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### weakly convergent subsequence implies strongly convergent

Statement: Let $X$ be a Banach space If $x_n \rightarrow x$ weakly and every subsequence of $\{x_n\}$ has a strongly convergent subsequence, then $x_n\rightarrow x$ strongly in $X$ Attempt: ?
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### Expand a function in Maclaurin's series

Please help me expand this function in Maclaurin's series to find an interval of convergence. I tried to turn it into this expression: but it doesn't give me the result. Help me please it's ...
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### Absolute and conditional convergence of function series

I have a problem. I have to explore absolute and conditional convergence of this function series Unfortunately. I didn't find in my reference any words about "absolute and conditional", Instead ...
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### Find a radius of convergence of power series

I have to Find a radius of convergence of this power series I' ve decided to use D'alambert indication: Looking for a limit i meet a problem with a factorial Please. help me finish this ...
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### explore the convergence of series with ln(n)

Help me explore the convergence of red color rounded series. On this photo (the equation below) I used radical indication but it doesn't show me the result. What would be better to use? ...
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### Series $\sum_{n=1}^{\infty}\frac{1}{(1+n)^{-z}} \$, $z \in \mathbb{C}$

I'm studying the series $$\sum_{n=1}^{\infty}\frac{1}{(1+n)^{-z}}$$ If $z = x+iy$, what is the behaviour of the series for $-1<x<0 \$?
### Problem on $L^2$ spaces.
Let $f_n$ be sequence of continuous functions on $[0,1]$ converging uniformly to $f$ a.e. on a set of finite measure. I would like to prove that this implies $f_n\rightarrow f$ in the $L^2$ norm. ...
### Relation between convergences in $L^{p}$ for probability spaces.
I have read that for a probability space $(\Omega,\Sigma,P)$ it is true that $f \in L^{p}(\Omega,\Sigma,P)$ implies $f \in L^{q}(\Omega,\Sigma,P)$ if $p>q$, and hence $L^{2} \subset L^{1}$. I'm ...