# Tagged Questions

For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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### Show that for almost all $x$ in $[-1,1]$, the series $\sum\limits_{n=1}^\infty \frac{1}{2^n \sqrt{|x-r_n|}}$ converges

Let $\{r_n\}$ be a sequence of real numbers in $[-1, 1]$, then show that for almost all $x$ in $[-1,1]$, the series $$\sum_{n=1}^\infty \frac{1}{2^n \sqrt{|x-r_n|}}$$ converges. I am struggling on ...
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### Statistics for Sports League Qualification

How can one quantify and predict the needed points for qualify in a league given an up-to-date results registry? For instance, regarding Basketball Euroleague, there's 8 teams in a league with direct ...
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### Closed form representation of an exponential series

Let $a\in\mathbb{R}$, $t\in\mathbb{R}\ge 0$ and consider the following series $$f(t)=\sum_{n=1}^{\infty}\frac{\exp({-a^2 t})-\exp({-n^2 t})}{n^2-a^2} .$$ The series is well defined in the poles at ...
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### Asymptotic Expansion for Function with an Embedded Integral [duplicate]

So I'm trying to find the asymptotic expansion as $x \to \infty$ of: $$f(x)=\frac{1}{\bigg[A-\int_{x_0}^x\frac{\lambda^y}{\Gamma(y+1)}dy\bigg]^{\frac{1}{\alpha}}}$$ where $x_0>0$ and $\alpha>0$ ...
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### Convergence of $\sum_{n=1}^\infty\frac{e^nn!}{n^n}$ [closed]

Is the following series convergent? $$\sum_{n=1}^\infty\frac{e^nn!}{n^n}$$
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### Prove that if $A$ is nonsingular, then the sequence $X_{k+1}=X_k+X_k(I-AX_k)$ converges to $A^{-1}$ if and only if $ρ(I-X_0A)<1$. [closed]

Prove that if $A$ is nonsingular, then the sequence $X_{k+1}=X_k+X_k(I-AX_k)$ where $A$ and $X_k$ are $n\times n$ matrices with $k=0,1,2,...$ converges to $A^{-1}$ if and only if $ρ(I-X_0A)<1$. I'...
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### limit of $|n^t\sin n|$

It is known that $\{\sin n : n\in\mathbb{N}\}$ is dense in $[-1,1]$, hence $\lim_{n\to\infty}\sin n$ doesn't exist and also $\lim_{n\to\infty} n^t\sin n$ doesn't exist for all $t>0$ (the reason is ...
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### Partial fraction decomposition: $f(x) = \frac{x}{1-x-x^2}$

Consider the function $$f(x) = \frac{x}{1-x-x^2}$$ (a) Determine a recursive formula for the coefficients $c_n$ of the Maclaurin series of $f$. (b)Using the partial fractions decomposition of $f(x)$...
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### Doubt on limit of sum of sequences. Two procedures leads to different answers.

I have the following problem. Determine the convergence or divergence of the sequence $(x_n)$ where $$x_n=\frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{2n}.$$ My first approach was: Well, ...
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### Confused about the choice of $\epsilon$ for proving sequences are not null

Definition of null sequence I am using: $(a_n) \to 0$ as $n \to \infty \iff$ given $\epsilon > 0$, there's $N$ such that $n > N \implies |a_n| < \epsilon.$ The sequences below are not null ...
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### How to show that if $f_{k}(x)=kxe^{-kx}$, {$f_k$} converges to the zero function?

$f_{k}(x)=kxe^{-kx}$ for all $x\ge0$ and $k\ge1$, how to show {$f_k$} converges to the zero function on $[0,\infty)$ pointwise? By L'hospital's Rule we can show that: $\lim_{x\to\infty}xe^{-x}=0$. ...
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### Fundamental theorem of arithmetic, prove that these matrices are the same apart from the main diagonal.

I know and I can prove that the fundamental theorem of arithmetic is encoded uniquely by this recurrence written in Mathematica 8: ...
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### Prove that $n^{n+1} \leq (n+1)^{n} \sqrt[n]{n!}$

Let $n$ be a positive integer. I conjectured that the following inequality is true $$n^{n+1} \leq (n+1)^{n} \sqrt[n]{n!} .$$ Anyhow I could neither prove nor disprove it. I ...
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### A problem involving the product $\prod_{k=1}^{n} k^{\mu(k)}$, where $\mu$ denotes the Möbius function

Let $\mu$ denote the Möbius function whereby \mu(k) = \begin{cases} 0 & \text{if $k$ has one or more repeated prime factors} \\ 1 & \text{if $k=1$} \\ (-1)^j & \...
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### Meaning of the numbers in a sequence definition

The sequence $(a_n)$ tends to $+ \infty \iff$ given any number $C$, there's a number $N$ such that $n > N \implies a_n \ge C.$ Given a certain $N$ it's not difficult to prove the implication, but ...
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### limit of sum with binomial coefficients

I have the problem to compute next double sum $$\sum_{n=2}^{\infty}\frac{(-1)^n}{n-1}\sum_{k=0}^{n}(3n-k)^j{n\choose k}A^{n-k}B^k\;,$$ being $j\gg1$ an integer number and \$...
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### Which integer sequence starts with small elements, and stays there for a (really) long time, but eventually escapes the initial area?

Graphically, I am searching for something like this: The only additional requirement would be that the elements are defined by a closed formula or "simple" recursion, i.e. no definition by cases (...