# Tagged Questions

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### Limit of series where termwise convergence holds

Given $a_n\in\mathbb{R}$ such that $\sum_{n=1}^\infty |a_n|^2<\infty$. Is it true that $$\lim_{r\rightarrow 1^-}\sum_{n=1}^\infty(1-r^n)^2|a_n|^2=0?$$ It is certainly true the limit of each term ...
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### Showing a sequence convergence

Let $a_1,a_2>0$ and $a_{n+1}=\cfrac{2}{a_{n-1}+a_{n}}(n\ge2)$, How to prove $a_n$ is convergent?
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### Let $G_n$ denote the geometric mean of binomial co-efficients… [duplicate]

Denote by $G_n$ the geometric mean of the binomial co-efficients $${n\choose 0},{n\choose 1},\ldots ,{n\choose n}$$ Prove that $$\lim_{n\to \infty}\sqrt[n]{G_n}=\sqrt e$$ My work: We have, ...
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### Calculate $\lim_{n \to \infty}U_n$

How can I calculate $\lim_{n \to \infty} U_n$ where $$U_n = \sum_{k=1}^n k \sin\left(\frac{1}{k} \right)?$$
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### Limit of $a_{n+1}= (2-a_n)a_n$

Let $(a_n)_{n \in\ \mathbb{N}}$ be a recursive sequence with $a_0= \frac{1}{2}$ and $a_{n+1}= (2-a_n)a_n ~~n \in\mathbb{N}$ (a) Show that $0 < a_n<1$ for all $n \in\mathbb{N}$ (b) Show that ...
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### Which points are limit points and which are isolated points?

Let $S = \left\{ x \in \mathbb{R}^{N} | \ ||x||_{2} \lt 1 \right\}$ $\cup$ $\left\{ x = (\zeta_{1},...,\zeta_{N})^{T} \in \mathbb{R}^{N} \ | \ ||x||_{2} = 1, \ \zeta_{1} \gt 0 \right\}$ Which points ...
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### Convergence proof that involves integral

For $n\in N$ let $a_n = \displaystyle \int_1^n \frac{\cos x}{x^2} dx$. Prove, for $m \geq n \geq 1$, that $|a_m - a_n| \leq n^{-1}$ and deduce that $(a_n)$ converges. By integration by parts, ...
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### Confusion on a problem on limit

Is it true that $$lim_{n->\infty}(1-(0.75)^n)^{2^n} = 0$$ and $$lim_{n->\infty}(1-(0.25)^n)^{2^n} = 1$$. Why ?