# Tagged Questions

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### Compute infinite sum of a arithmetico-geometric series $\sum_{i=0}^{\infty} \frac{i}{2^i}$ [duplicate]

I am trying to compute the sum $\sum_{i=0}^{\infty} \frac{i}{2^i}$ which I know should be equal to $2$, but I cannot prove it. If I am not mistaken, it should be a arithmetico-geometric series ...
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### Does series converge or not?

$$\sum_{n=1}^\infty~\left|\frac{\cos2^n}{n}\right|$$ I just confused what to do.
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### Alternating p series. given that summation

Given that $$\sum_{k=1}^\infty{\frac{1}{k^2}} = \frac{\pi^2}{6}\$$ Show that $$\sum_{k=1}^\infty{\frac{(-1)^{k+1}}{k^2}} = \frac{\pi^2}{12}\$$
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### A question about limsup and limif

Could you please help me understand this question: Suppose $a_n$ is bounded sequence and $A<\liminf a_n$, $B>\limsup a_n$. Prove : $A<a_n<B$ for all n>N. It seems to me to simple to be ...
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### Calculate the following sequence $\sum_{n=0}^{+\infty }\left ( -\dfrac{1}{4\alpha } \right )^{n}\dfrac{ (2n)!}{n!},\; \alpha >0$

Calculate the following sequence $$\sum_{n=0}^{+\infty }\left ( -\dfrac{1}{4\alpha } \right )^{n}\dfrac{ (2n)!}{n!},\; \alpha >0$$
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### Sequence bounds and limit

I'm doing the following exercise. Given the sequence \begin{cases} a_{n+1} = {n + 8\over4n + 1}*a_n & n=0, 1, 2 \\ a_{0} = 1 \end{cases} Find if the sequence is definitely decreasing/increasing. ...
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### Convergance of a sequence [on hold]

Prove that the sequence $(a_n)$ converges, where$$a_n=\frac {3+n+4{n^2}}{1-n+3{n^2}}$$ for all $n\ge1$
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### Multiplying non-decreasing sequences

Let $(a_n)$ and $(b_n)$ be non-decreasing sequences of positive terms (i.e. $a_n\gt0$ and $b_n\gt0$ for all $n\ge1$). Prove that the sequence $(c_n)$ is non-decreasing, where $c_n=a_nb_n$ for all ...
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### Problem with Sequences..

I'm having trouble with a homework question, and cant see where im going wrong. The question is as follows: Let $x\gt1$ and let $a_n = 1+\frac 1x+\frac 1{x^2} + \cdots + \frac 1{x^n}$ for all ...
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### suming an infinite series

the question involves person A eating half a loaf of bread. then person B eats half of the half left over. then person A eats half of whats left over... etc. I defined the series of person A as ...
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### Finding the limit of a sequence by diagonalising a matrix

Consider the sequence described by: $\frac11 , \frac32 , \frac75 , ... ,\frac {a_{n}}{b_{n}}$ where $a_{n+1} = a_n +2b_n$ and $b_{n+1} = a_n+b_n$ Find a matrix $A$ such that ...
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### How to prove convergence of $a_n$ if $(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$?

Could you give me some hint how to conclude convergence of $a_n$ from this feature : $$(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$$ From $$(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$$ we may conclude that ...
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### Construct a Converging Series from the Following

This is more of a request for advice than a request for solution. Last night we were given the following and nobody figured it out in the time given (about 5 minutes). I think this is a problem many ...
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### Determine if the given sequence converges or diverges

Let $(x_n)$ be a sequence defined as $x_n = \frac{1}{n} \sum_{j=1}^{n} \frac{j+1}{j^2}$ . We want to know if $(x_n)$ converges. The trouble I am having here is that the sum depends on $n$. We know the ...
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### Determine if $\sum\limits_{n=1}^{\infty}(1+\dfrac{2}{n})^n$ converges or diverges

I have an infinite series $\sum\limits_{n=1}^{\infty}(1+\frac{2}{n})^n$. I need to show if it converges or diverges using any test. I've tried applying all of the tests that I know, and hit dead ...
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### A question about using Squeeze Theorem to solve theoretical convergence question

Could you give me some hint how to deal with this question: Suppose $a_n\le b_n \le c_n$ for almost all n, $b_n\to L$, $c_n-a_n\to 0$. Prove: $a_n \to L,b_n \to L$. Well, if $a_n\to a, b_n \to b$ ...
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### Show that $\sum\limits_{n=1}^\infty\dfrac{2n^2-1}{3n^5+2n+1}$ converges or diverges

I'm working with some infinite series problems and I have to show that the series $\sum\limits_{n=1}^\infty\dfrac{2n^2-1}{3n^5+2n+1}$ converges or diverges. I don't have a lot of experience doing ...
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### Does $\sin(\sin(\sin\cdots(\sin1)\cdots) \rightarrow 0$?

Stuck on homework problem (not this), if I can prove as a lemma that the sequence $$\sin(\sin(\sin\cdots(\sin1)\cdots) \rightarrow 0$$ then I'm done. It's monotonic and decreasing and bounded by 0 ...
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### Find a general formula for x_k

The sequence $x_k$... is defined by $x_0 = 0, x_1 = 2$, and $x_{k+2} = 6x_{k+1}−13x_k$ for $k≥0$. Find a general formula for $x_k$. I actually came here because I found a solution on here for a ...
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### The summation of product of factorials

So the question is $\sum\limits_{x=0}^n \frac{(\beta+n-x)! (\alpha+x)!}{x!(n-x)!}$. I got the following result from mathematica yet I don't know how to prove it. Can anyone give me some help?
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### Superlinearly convergent

A sequence $\{p_n\}$ is said to be superlinearly convergent to $p$ if $$\lim_{n\to \infty}{\frac{|p_{n+1}-p|}{|p_n-p|}}=0$$ a. Show that if $p_n\to p$ of order $\alpha$ for $\alpha>1$, then ...
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### Question on Morse inequalities

I want to understand why: if i have then $(4.1)$ is formal : it means that please help me Thank you EDIT1: $(4.1)$ tel us that $\displaystyle\sum_{q=0}^{\infty} (M_q-\beta_q)t^q=(1+t)Q(t)$ ...
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### Short question on power series

After applying ratio test, the result L = x^2/4, n approaches infinity. So when you're trying to find the radius of convergence of: $$x^2/4<1$$ is it $$x<2?$$ Therefore radius is 2? I want ...
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### $\sqrt[\infty]{\infty^2}$ in limit of series using root test

I'm trying to solve a problem to show if the infinite series $\sum\limits_{k=1}^{\infty}\dfrac{k^2}{2^k}$ converges or diverges using the root test. When put in limit form, I got ...
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### Infinite series: which test

I'm having troubling deciding which test to use for this: $$\sum_{n=1}^\infty (-1)^n\arctan\left(\frac{\ln(n!)}{n+4^n}\right)$$ I tried altnerating test and ratio test but I couldn't get an answer. ...
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### Infinite Series with factorial

I'm having trouble manipulating the function of this series which has factorials to show that it converges or diverges using the ratio test. The series is ...
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### How to prove it?

Let $y_0\geqslant 2$, $y_n=y_{n-1}^2-2$, $n\in\mathbb{N}_+$, set $\displaystyle S_n=\sum_{k=0}^{n}\frac{1}{y_0\cdots y_k}$, how to prove $$\lim_{n\to\infty}S_n=\frac{y_0-\sqrt{y_0^2-4}}{2}.$$ Do you ...
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### What is the main defferences between nets and ordinary sequences

I know that there are many results in metric spaces (or first-countable topological spaces) can be describe in the language of sequences but these results might not be true in general topological ...
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### Use the squeezing theorem to find the limit of the sequence

Is anyone able to help me answer this question? Or point me in the right direction? Use the squeezing theorem to find the limit of the sequence $\{a_n\}_{n=1}^{\infty}$ with $n$-th term ...
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