# Tagged Questions

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### On the convergence of a series! [on hold]

Suppose that $a_n>0$ and $\sum_{n=1}^{\infty}a_n$ converges. Prove that: $$\sum_{n=1}^{\infty}\left(\frac {a_n}{\ln(1+n)}\ln\frac 1{a_n}\right)$$ Converges too! Reference: Titu Andrescu-Problems in ...
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### Proving an inequality for a sequence by induction

I'm having some trouble with the following problem: Let $a_n$ be a sequence defined iteratively for $n \geq 0$ as follows: $a_n = a_{m+1} + 2a_m + a_{n-m-1} + 2$ where $m$ is defined as ...
1answer
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### Inequality in non-decreasing sequence

Let $a, b$ be two sequences of real numbers such that $a_1 \le a_2 \le \dots \le a_n$ and $b_1 \le b_2 \le \dots \le b_n$. Prove (or disprove) that ...
1answer
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### Prove an inequality

for $x > -1$, Prove the following inequality: $$\left( {\ln (1 + x) + \sum\limits_{k = 1}^n {\frac{{{{( - 1)}^k}{x^k}}}{k}} } \right){( - x)^{n + 1}} \le 0$$ Following the advice to use ...
1answer
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### Find an N satisfying this inequality

I am trying to catch up again. $$\frac{1}{2^{n}} + \frac{1}{3^{n}} + \frac{1}{4^{n}} < \frac{1}{365}$$ Find an $N$ whereby all $n \geq N$ give correct outcomes I thought that as long as ...
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### Show that $\lim \inf a_n\le\lim\inf s_n.$

Let $\{a_n\}$ be a bounded sequence of real numbers. Let $s_n=\dfrac{a_1+a_2+\cdots+a_n}{n}~\forall~n\in\mathbb N.$ Show that $\lim \inf a_n\le\lim\inf s_n.$ The only definition I know of limit ...
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### Convergence of the sequence $v_n$ governed by $$v_{n+1}^2\le (1+\alpha_n)v_n^2+\gamma_nv_n-u_n+\beta_n,\quad n\ge 1$$

My question is that if there is a sequence of positive reals $v_n$ which is governed by the following inequalities $$v_{n+1}^2\le (1+\alpha_n)v_n^2+\gamma_nv_n-u_n+\beta_n$$ where ...
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### Inequalities and sequences

I ran across this problem in Gelfands Algebra today. 1/2 < 1/101 + 1/102 + ... + 1/200 < 1 He shows proof that this is true 1/2 < 1 - 1/2 + 1/3 - 1/4 + ... + 1/199 - 1/200 < 1 Again he ...
1answer
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### the best constant in an inequality?

I learnt how to show the below inequality by C-S inequality: k is from $0$ to $\infty$ If $\sum a_{k}^{2}9^{k}\le 5$ then $\sum |a_{k}|2^{k}\le 3$. next,I tried to show that 3 is the best possible ...
1answer
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### How prove this $\frac{1}{b_{k+1}b_{k+2}}+\frac{1}{b_{k+2}b_{k+3}}+\cdots+\frac{1}{b_{2k}b_{2k+1}}>\frac{1}{12345},$

let sequence $\{b_{n}\}$,and $b_{n}>0$,let $$S_{n}=b_{1}+b_{2}+\cdots+b_{n}\le n^{\frac{3}{2}},\forall n\ge 1$$ show that ...
3answers
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### Prove that $\,\,\displaystyle\inf_{n\in\mathbb N}\sum_{k=0}^{p}\lvert\sin{(n+k)^p}\rvert>0$

For any postive integer number $p$, show that $$\inf\left({\vert\sin{(n^p)}\vert+\vert\sin{(n+1)^p}\vert+\cdots+\vert\sin{(n+p)^p}\vert,\,n\in \mathbb{N}}\right)>0$$ My try: I only can prove by ...
0answers
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### Increasing fraction of sum of binomial coeffients

Let $n$ be a positive integer. Show that the quantity $$\displaystyle \frac{ \displaystyle \sum_{i=1}^n { n+k \choose i-1 } }{ \displaystyle \sum_{i=1}^n { n+k+1 \choose i } }$$ is ...
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### Showing $|\sum_{k=1}^n\frac{\sin(kx)}k|<2\sqrt{\pi}$

For any real $x$ and positive integer $n$, is it true that: $$\left|\sum_{k=1}^n\frac{\sin(kx)}k\right|<2\sqrt{\pi}\quad ?$$ Please justify.
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1answer
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### Proving a lim sup inequality in Chapter 12 of Kenneth Ross's “Elementary Analysis”

I am struggling to digest and understand Theorem 12.2 (p. 76) in Elementary Analysis: The Theory of Calculus by Kenneth Ross Theorem 12.2 Let $s_n$ be any sequence of nonzero real numbers. ...
1answer
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### The completeness of the real numbers with respects to Cauchy sequences?

My Question: I am not sure about the very last inequality in the proof below; namely, where did we get $\mid a_{n}-a_{N}\mid$ and $\mid a_{N}-b\mid$? I see that $\mid a_{n}-a_{N}\mid<\epsilon/2$ ...
2answers
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### Solving for $n$ in a geomtric progression

Given the general term of geometric sequence: $a_n = \dfrac{x}{2^n}$ I would like to solve for the value of n that makes $a_n =1$. My work so far: \begin{align*} a_n &= \frac{x}{2^n}\\ 2^n &= ...