# Tagged Questions

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### In the definition of sequences diverging to infinity, why must the constants be positive?

We are given the following definitions A sequence $(a_n)$ diverges to $\infty$ if for each $M \in \mathbb{R}^+ \exists N_M \in \mathbb{N} \ \text{such that } \\ a_n > M \ \forall n \geq N_M$ ...
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### Limit of sequences: $\lim \frac{(2n)!}{(n!)^2}$

Verify if the sequence $$\frac{(2n)!}{(n!)^2}$$ converges. My attempt: $$\frac{(2n)!}{(n!)^2} = \frac{(2n)(2n-1)...(n+1)}{n.(n-1)...1} \geq \frac{(n+1)^n}{n!}$$ Maybe it is easier to show that ...
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### Proving that $\sum \frac{n^{n+1/n}}{(n+1/n)^n}$ diverges

Show that the series $$\sum \frac{n^{n+1/n}}{(n+1/n)^n}$$ diverges The ratio test is inconclusive and this limit is not easy to calculate. So I've tried the comparison test without success.
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### $\sum r^n |\sin(nx)|$ convergence

Verify if the series $$\sum r^n |\sin(nx)|,\qquad r>0$$ Converges or diverges I've tried some comparisons with known series and the convergence tests, but didn't work. I think we ...
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### convergence of a sequence of cauchy

I would like to ask something about the convergence of a Cauchy sequence in a space $X$ metric. There will be a metric space $X$ such that If $(x_n)$ cauchy sequence in $X$ then $(x_n)$ is not ...
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### necessary condition on the sequence $\{a_n\}$ and $\{b_n\}$ such that $\sum a_n b_n$ converges [on hold]

Is there any necessary condition on the sequence $\{a_n\}$ and $\{b_n\}$ such that $\sum a_n b_n$ converges ?
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### Viewing a sequence as a function on the space of positive integers

I see the following lines in a book : " Consider a bounded sequence of real or complex numbers $\{\eta_n\}$. Such a sequence $\{\eta_n\}$ defines a function $x(n) = \{\eta_n\}$ defined on the ...
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### Limit of sum of series

Let $$\sum_{i=1}^\infty c_i^{(0)}$$ be a positive convergent series. Then the series: $$\sum_{i=1}^\infty c_i^{(1)}=\sum_{i=1}^\infty \ln(1+c_i^{(0)})$$ is also a positive convergent series. ...
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### Sequence converging to the supremum. [on hold]

let $T\subset \mathbb R$ let $a=\sup(T)$, and suppose that and $a<\infty$ and that $a \notin T$. Show that there exist a sequence $(a_n)\subset T$ for which $\lim_{n\to\infty}a_n=a$. If you are ...
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### A identity relating a infinite series and a definite integral [duplicate]

Prove that, $$\sum_{n=1}^{\infty} \frac{1}{n^n} = \int_{0}^{1} x^{-x}dx$$ I made no significant progress, I'm looking for hint/ideas to approach this problem. Thanks!
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### Identifying or bounding the zeros of the composition of two generating functions

Given two generating functions $$G(a_n;x)=\sum_{n=0}^\infty a_nx^n \quad\text{ and }\quad H(b_n;x)=\sum_{n=0}^\infty b_nx^n,$$ what techniques are available for locating, or finding bounds on, the ...
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### Question about prove of Absolute Convergence Test?

I'm self-studying from the book Understanding Analysis by Stephen Abbott and I have a question about the proof of the Absolute Convergence Test (theorem 2.7.6 on page 65). The author states that ...
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### Relation between limit of functions and sequences

I need to prove that the sequence $$\frac{\sqrt{n}}{\log n}$$ diverges. I know that $$\lim \frac{\sqrt{x}}{\log x} = \infty$$. Is there any theorem that relations the limit of the function with the ...
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### Two sequences defined by recurrence relations satisfy $x_n/y_n<\sqrt{7}$ for all $n$

Let $(x_n)_{n\geq 1}$ and $(y_n)_{n\geq 1}$ be two sequences such that: $$x_{n+1}=x_n^2+1 \quad \text{ and } \quad y_{n+1}=x_n y_n$$ with $x_1=2$ and $y_1=1$ Prove that for all $n$ ...
So I have the real sequence for fixed $x\in \mathbb{R}$: $y_{j}(x)=\begin{cases}f_{j-1}(x) &\text{if } |f_{j-1}(x)|\leq |x_{j}|, \\ f_{j}(x) & \text{else}, \end{cases}$ where 1) ...