# Tagged Questions

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### Prove or disprove that there exists a unique positive integer sequence $\{a_{n}\}$ satisfying a condition

Question: Prove or disprove: there exists a unique positive integer sequence $\{a_{n}\}$ satisfying the following condition: $\forall m\in N^{+}$, there exists a unique integer sequence ...
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### Initial value for a sequence to become periodic.

The following is from the previous Proofathon contest: Let $a_{n}$ be the sequence defined by the recursion $\sqrt{a_{n+1}}= (2(\sqrt[2014]{a_n})-1)^{2014}.$ Find all the values of $a_1$ ...
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### How can I define a “formula” for general term of a sequence with some given values?

I have a doubt: If I have $\alpha, \beta, \gamma, \delta$ natural numbers, how can I write a formula to generate infinite sequences, such that $f(1)=\alpha, f(2)=\beta, f(3)=\gamma, f(4)=\delta$? I ...
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### Infinite Sum of products

What is the infinite sum $$S = {1 + \frac{1}{3} + \frac{1\cdot 3}{3\cdot 6} + \frac{1\cdot 3\cdot 5}{3\cdot 6\cdot 9}+ ....}?$$ I attempted messing around with the $n$ th term in the series but didnt ...
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### Series Summation involving factorials, and powers.

What is the value of $\dfrac{1.2}{3!} + \dfrac{2.2^2}{4!} + \dfrac{3.2^3}{5!} + ...... + \dfrac{15.2^{15}}{17!}$ How would you proceed with this? I attempted writing the general term and tried some ...
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### Difficult infinite trigonometric series

Evaluate the sum of the following infinite series. $$\left(\sin{\frac{\pi}{3}}\right) + \left(\frac{1}{2}\sin{\frac{2\pi}{3}}\right) + \left(\frac{1}{3}\sin{\frac{3\pi}{3}}\right) + \ldots$$
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### Suppose for all $n$, $a_{n+1}\le a_n + \frac1{n^p}$. Find all positive $p$ such that we can guarantee $\{a_n\}$ always converge.

Let $\{a_n\}$ be any sequence of positive real numbers. Suppose for all $n$, $a_{n+1}\le a_n + \frac1{n^p}$. Find all positive $p$ such that we can guarantee $\{a_n\}$ always converge. For example, ...
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### Prove $1 + \sum_{i=0}^n(\frac1{x_i}\prod_{j\neq i}(1+\frac1{x_j-x_i}))=\prod_{i=0}^n(1+\frac1{x_i})$

Prove the identity $$1 + \sum_{i=0}^n \left(\frac1{x_i}\prod_{j\neq i} \left(1+\frac1{x_j-x_i} \right) \right)=\prod_{i=0}^n \left(1+\frac1{x_i} \right)$$ and hence deduce the inequality in Problem ...
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### Compute $\sum_{j=1}^k\cos^n(j\pi/k)\sin(nj\pi/k)$

Compute the series $\sum_{j=1}^k\cos^n(j\pi/k)\sin(nj\pi/k)$ Hint: the answer is in fact 0
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### These two sequences have the same limit

Let $a_1$ and $b_1$ be any two positive numbers, and define $\{ a_n\}$ and $\{ b_n\}$ by $$a_n = \frac{2a_{n-1}b_{n-1}}{a_{n-1}+b_{n-1}},$$ $$b_n = \sqrt{a_{n-1}b_{n-1} }.$$ Prove that the ...
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### Probability that the first digit of $2^{n}$ is 1

Let $a_{n}$ be the number of terms in the sequence $2^{1},2^{2},\cdots ,2^{n}$ which begins with digit 1. Prove that $$\log2 -\frac{1}{n}<\frac{a_{n}}{n}<\log2\text{ (log base is 10)}$$ ...
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### Summation of a finite series

Let $$f(n) = \frac{1}{1} + \frac{1}{2} + \frac{1}{3}+ \frac{1}{4}+...+ \frac{1}{2^n-1} \forall \ n \ \epsilon \ \mathbb{N}$$ If it cannot be summed , are there any approximations to the series ?
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### Is $u_n\le(1-a)^n\forall n\in\mathbb{N}$?

Consider the sequence $\{u_n\}$ where $u_0=1,u_1=1-a$ for some $0< a < 1/4$, and $u_{n+2} = u_{n+1}-au_n$. Is $u_n\le(1-a)^n\forall n\in\mathbb{N}$?
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### how prove $\sum_{n=1}^\infty\frac{a_n}{b_n+a_n}$is convergent?

Let$a_n,b_n\in\mathbb R$ and $(a_n+b_n)b_n\neq 0\quad \forall n\in \mathbb{N}$. The series $\sum_{n=1}^\infty\frac{a_n}{b_n}$ and $\sum_{n=1}^\infty(\frac{a_n}{b_n})^2$ are convergent. How to prove ...
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### $\limsup\left(\frac{a_1+a_{n+1}}{a_n}\right)^n\ge c$

Let $a_n>0,n\in\mathbb{N}$ be a sequence of positive real numbers. There exists a positive real number $c$ such that $\limsup\left(\frac{a_1+a_{n+1}}{a_n}\right)^n\ge c$ as $n\to\infty$ for all ...
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### Convergence of series with modified denominator

Suppose the series with positive terms $\sum_{n=1}^{\infty} a_n$ converges. Let $r_n=\sum_{k=n}^{\infty}a_k$. Prove or disprove that $\sum_{n=1}^{\infty}\frac{a_n}{r_n}$ diverges, and prove or ...
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### Does there exist a sequence of real numbers $\{a_n\}$ such that $\sum_na_n^k$ converges for $k=1$ but diverges for every other odd positive integer?

Does there exist a sequence of real numbers $\{a_n\}$ such that $\sum_na_n^k$ converges for $k=1$ but diverges for every other odd positive integer?
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### How to compute the series $\sum\limits_{x=0}^\infty\sum\limits_{y=0}^\infty\sum\limits_{z=0}^\infty\frac{1}{2^x(2^{x+y}+2^{x+z}+2^{z+y})}$

How to compute the series $\displaystyle\sum_{x=0}^\infty\sum_{y=0}^\infty\sum_{z=0}^\infty\frac{1}{2^x(2^{x+y}+2^{x+z}+2^{z+y})}$ ? Thanks in advance.
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### * if $\sum_{n=1}^\infty u_{n}$ be divergent then $\sum_{n=1}^\infty n u_{n}$ is convergent or divergent *

let $\sum_{n=1}^\infty u_{n}$ be divergent $\sum_{n=1}^\infty n u_{n}$ this series is divergent or convergent? thanks in advance
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### A question about series with a strange property.

Does there exist a sequence $\left(a_n\right)_{n\ge1}$ with $a_n < a_{n+1}+a_{n^2}, \forall n=1,2,3,\ldots$ such that the series $\displaystyle{\sum_{n=1}^{\infty}a_n}$ converges? This is the ...
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### Sequence $a_k=1-\frac{\lambda^2}{4a_{k-1}},\ k=2,3,\ldots,n$.

Consider the sequence $a_1, a_2,\ldots,a_n$ with $a_1=1$ and defined recursively by $$a_k=1-\frac{\lambda^2}{4a_{k-1}},\ k=2,3,\ldots,n.$$ Find $\lambda>1$ such that $a_n=0$. The answer is ...
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### Does there exist a sequence $\{a_n\}_{n\ge1}$ with $a_n < a_{n+1}+a_{n^2}$ such that $\sum_{n=1}^{\infty}a_n$ converges?

Does there exist a sequence $\{a_n\}_{n\ge1}$ with $a_n < a_{n+1}+a_{n^2}$ such that $\sum_{n=1}^{\infty}a_n$ converges? Does there exist a sequence with the same property but with each term ...
### Evaluate the sum $\sum_{k=0}^{\infty}\frac{1}{(4k+1)(4k+2)(4k+3)(4k+4)}$?
Evaluate the series $$\sum_{k=0}^{\infty}\frac{1}{(4k+1)(4k+2)(4k+3)(4k+4)}=?$$ Can you help me ? This is a past contest problem.