# Tagged Questions

Convergence of sequences and different modes of convergence.

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### How to find Taylor series when $x_0=0$ and radius of convergence for $\frac{x}{1+x}$ for $f:(-1,\infty)$

Through the taylor series formula: $$f(x)=f(x_0)+f'(x_0)(x-x_0)+\frac{f''(x_0)}{2!}(x-x_0)^2+\dots+\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n$$ I've got that $f(x)=x-x^2+x^3-x^3\dots$ however my teacher claimed ...
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### finding the radius of convergence of $\sum_{n=1}^{\infty} n^2x^n$ [closed]

How does one find the radius of convergence of: $\sum_{n=1}^{\infty} n^2x^n$ using the fact that it's possible to differentiate every term. I have no idea how to go about with this
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### Divergence of $\sum \limits_{n=1}^{\infty}\frac{a_n}{a_1+a_2+\dots+a_n}$ [duplicate]

Suppose that $\sum \limits_{n=1}^{\infty}a_n$ series with positive terms which diverges then series $\sum \limits_{n=1}^{\infty}\dfrac{a_n}{a_1+a_2+\dots+a_n}$ also diverges. Can anyone show how to ...
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### Is this series divergent or convergent?

Please explain what method you used to prove so. $$\sum_{n=3}^\infty \frac{\tan\left(\frac{\pi}{n}\right)}{n}$$
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### Does this sequence of random variables converge almost surely?

I was trying to understand why almost sure convergence doesn't imply convergence of the mean and I encountered this answer. However, I do not understand why this sequence of random variables ...
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I have the following statement - The Taylor series of $\frac{x}{x+2}$ around $X = 1$ has a radius of convergence of $R = 4$. Is it right to say that this statement is false because a function is ...
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### Prob. 4 (b), Sec. 20 in Munkres' TOPOLOGY, 2nd ed: Which of these sequences are convergent w.r.t. the product, uniform, and box topologies?

Let $\mathbb{R}^\omega$ denote the set of all the (infinite) sequences of real numbers. Then which of the following sequences in $\mathbb{R}^\omega$ are convergent (and if so, then to which points(s)) ...
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### Finding convergence zone/range for $\sum_{i=1}^\infty \frac{x^{n^2}}{n(n+1)}$

$$\sum_{i=1}^\infty \frac{x^{n^2}}{n(n+1)}$$ I used the ratio test and I end up with: $$|x|*\frac{n}{n+2}$$ What steps do I need to take to continue? Looking for hints or steps, not full solution/
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### Definition of convergent sequence in topological space

The book I'm reading says: If $\{x_n\}$ is a sequence of points in $(X,T)$ and $x \in (X,T)$, the sequence is said to converge to $x$ iff for each open set $U \in T$, there exists an $n_0 \in \Bbb N$ ...
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### Sequence Cauchy w.r.t $L^2$-norm but has no convergent subsequence

I have a problem with following exercise: Let $C([0,1])$ be the space of continuous real valued functions on the closed unit inverval $[0,1]$. And let the $L^2$-norm given by: \Vert f\...
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### radius of convergence of ${3^{k^2}}{x^{k^2}}$
find the interval of convergence $\Sigma _{k=0} ^{\infty} 3^{k^{2}} x^{k^{2}}$ The radius of convergence of this series is 1/3 by the book. but the answer and what i think is so different what i ...
### Does $\lVert S_Nf-f\rVert_{\infty}\to 0$ imply $\lVert S_Nf-f\rVert_{L^2}\to 0$?
Does $\lVert S_Nf-f\rVert_{\infty}\to 0$ imply $\lVert S_Nf-f\rVert_{L^2}\to 0$ I have a proof for the first case, under the assumption that $f$ is $C^1$ and real valued (also $1$ periodic) \$\lVert ...