# Tagged Questions

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### Question about limit points in relation with continuity and functional limits

I'm self-studying from the book Understanding Analysis by Stephen Abbott, and I have the feeling that the author is being careless about limit points in his theorems or I am not understanding ...
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### Uniform convergence of $xe^{-nx}$

Does the sequence $(f_n)$ on $[0, \infty)$ given by $f_n(x) = > xe^{-nx}$ converge uniformly? This is from Bartle's Elements of Real Analysis. I've already proven that the sequence is ...
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### Convergent or divergent $\sum_{n=1}^{\infty} \frac{e^nn!}{n^n}$?

Any suggestion/hint, not the whole solution, how to determine convergence/divergence of $$\sum_{n=1}^{\infty}\dfrac{e^n \cdot n!}{n^n}$$ I'm currently stuck.
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### Series representation of cosh or the hyperbolic cosine. [closed]

What is the Series representation of cosh or so-called the hyperbolic cosine? Can you help me guys?
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### Prerequisites for understanding G.H. Hardy's 'Divergent Series'

I picked up a copy of G.H. Hardy's 'Divergent Series' a few days ago. So far I love it, as I love the ideas associated with sequences and series, but I am finding it a bit difficult to understand. I ...
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### Divergent subsequence of an unbounded sequence

Let $(a_n)$ be a sequence of real numbers that is unbounded above. Show that $\exists$ a subsequence $(a_{n_k})_{k \ge 1}$ such that $\lim_{k \rightarrow \infty} a_{n_k} = + \infty$. Working ...
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### Proof Verification - Every sequence in $\Bbb R$ contains a monotone sub-sequence

Came across the following exercise in Bartle's Elements of Real Analysis. This is the solution I came up with. Would be grateful if someone could verify it for me and maybe suggest better/alternate ...
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### On Lucas Lehmer primality Test

http://primes.utm.edu/notes/proofs/LucasLehmer.html is proof of the Lucas Lehmer Test I read. The part I do not understand is why did he consider the sequence $S_n=S_{n+1}^2-2$. I mean why would ...
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### $a, z_1 \gt 0$ and $z_{n + 1} = (a + z_n)^{\frac 1 2}$ then $(z_n)$ is monotone and bounded?

Having trouble with the following exercise in Bartle's Elements of Real Analysis. Let $a, z_1 \gt 0$. Define $z_{n + 1} = (a + z_n)^{\frac 1 2}$ for $n \in \Bbb N$. Show that $(z_n)$ converges. ...
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### Absolute Convergent Series Laws (Exercise)

Me again, I have troubles to understand the concept of absolute convergence in the general setting when the set can be uncountable. In the book what I read says: Definition: Let $X$ be a set ...
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### Fubini's Theorem for Infinite series

In the book what I've read, there is one point where the author suggest to begin the proof of the Fubini's Theorem for infinite sum in the case when is non-negative after this try to generalize. But ...
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### Testing Boundary Points Of $\sum_{n = 1}^{\infty} \frac{n!z^n}{n^n}$

I'm having some trouble testing the series indicated in the title at its boundary points. I'll sketch the preliminary work, then arrive at the problem. It is clear that the series converges absolutely ...
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### Associativity of Finite Partial Sums in a Convergent Series

So I saw some topics on this, but they didn't seem to answer exactly what I was looking for. Self-learning through Understanding Analysis, a exercise is the following: So my misunderstanding is ...
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### Interchange of finite sum with convergence sequences.

Hi everyone I'm wondering if the following proof is correct (to be honest at the beginning I have some troubles to understand what the sequences of the sums of convergent sequences has to be, but I ...
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### Calculus: computation of $\sum \frac{2^i}{i!}$

$$\sum_{i=0}^\infty \frac{2^i}{i!}$$ Would anyone mind telling me what is the answer? I know this may be a silly question but I would like to know.
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### Summation of Modulo Sequences

I came across through simulation that multiplying and adding certain modulo sequences yield equal results. Consider the following two sequences \begin{align} g_0[k] &= \sum_{n=0}^{N-1} \left< ...
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### Proof for length of period in simple modulo $N$ sequence.

I am looking for a concise proof that the length of the smallest period of the sequence $$f[n] = a n \pmod N$$ is $N$ if $(a,N) = 1$. From the Pigeonhole Principle, it is not hard to show that ...