# Tagged Questions

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### Prove that for any integer 𝑛 ≥ 1, we have [on hold]

Prove that for any integer 𝑛 ≥ 1, we have
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### Which of the following is correct?

Let $$X = \frac{1}{1001} + \frac{1}{1002} + \frac{1}{1003} + \cdots + \frac{1}{3001}$$ Then find the correct option. (A) $X < 1$ (B) $X > \frac{3}{2}$ (C) $1 < X < \frac{3}{2}$ (D) ...
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### Finite Series Inequality [duplicate]

For each $n=1,2,{\dots}$ and $x\in(0,{\pi})$, prove that the series $$S_n(x)=\sum_{k=1}^{n} \frac{\sin(kx)}{k}>0$$
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### Binomial coefficient - first two terms, proof of inequality

I've seen the following and I'm not sure whether it is true or not, and if yes, why it holds. $(1-p)^x \geq 1-p x$ for $p\in (0,1)$ and $x>0$. Do I need some additional Information to prove ...
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Question: Consider the following sequence : $$a_1=1 ; a_n=\frac{a_{[\frac{n}{2}]}}{2}+\frac{a_{[\frac{n}{3}]}}{3}+\ldots+\frac{a_{[\frac{n}{n}]}}{n}$$. Prove that: $$a_{2n}< 2a_{n } (\forall ... 1answer 18 views ### Exercise on a series Prove the following inequality:$$\sum_{k=m+1}^{\infty} \frac{1}{k!}< \frac{1}{m*m!} \forall m\in \mathbb{N^+}$$My strategy of attack was to set up an inequality like ... 2answers 30 views ### sequence a_n = \lceil \sqrt{2}n \rceil  I was trying to prove \lceil \sqrt{2}n \rceil + \lceil \sqrt{2}m \rceil \geq \lceil \sqrt{2}(n+m) \rceil where m,n\in \mathbb{z} Direct proof I tried but could not figure out. I tried fixing m ... 3answers 119 views ### Showing n!<e(\frac{n}{2})^n I'd like to prove that n!<e(\frac{n}{2})^n. What I have so far: \sqrt[n]{n!} = \sqrt[n]{1\cdot 2 \cdot \ldots \cdot n} \leq \frac{1+\ldots +n}{n}=\frac{(n+1)n}{2n}=\frac{(n+1)}{2}. Thus ... 1answer 59 views ### Very complicated limit and trying to find convergence I have no idea how to prove this: \lim_{n \to \infty} \frac{2^2 \times4^2\times6^2\times\dots\times(2n)^2}{(1\times3)(3\times5)\dots((2n-1)(2n+1))} = \lim_{n \to \infty}\frac{2^2 ... 2answers 72 views ### e^x>1+(1+x)\log(1+x),for x>0 using infinite series. e^x>1+(1+x)\log(1+x),for x>0 using infinite series. Attempt \sum x^k(\frac{1}{k!}+\frac{(-1)^{k+1}}{k!}+\frac{(-1)^{k+1}x}{k!})>1. any hints? 1answer 59 views ### Inequality Question about Converging Sum This is from the UPenn prelim questions. http://hans.math.upenn.edu/amcs/AMCS/prelims/prelim_review.pdf (I asked the question before, and there was no answer, so I am asking it again.. I'm not sure ... 2answers 214 views ### Waring's Inequality Solution$$ \text{Waring's problem asks, "Is }\left\lfloor \left(\frac{3}{2}\right)^n\right\rfloor =\left\lfloor \frac{3^n-1}{2^n-1}\right\rfloor\text{ always true?"} $$We craft an inequality, with m,n ... 1answer 67 views ### Proving an inequality for a sequence by induction I'm having some trouble with the following problem: Let a_n be a sequence defined iteratively for n \geq 0 as follows: a_n = a_{m+1} + 2a_m + a_{n-m-1} + 2 where m is defined as ... 1answer 36 views ### Inequality in non-decreasing sequence Let a, b be two sequences of real numbers such that a_1 \le a_2 \le \dots \le a_n and b_1 \le b_2 \le \dots \le b_n. Prove (or disprove) that ... 1answer 53 views ### Prove an inequality for x > -1, Prove the following inequality:$$\left( {\ln (1 + x) + \sum\limits_{k = 1}^n {\frac{{{{( - 1)}^k}{x^k}}}{k}} } \right){( - x)^{n + 1}} \le 0$$Following the advice to use ... 1answer 63 views ### Find an N satisfying this inequality I am trying to catch up again.$$ \frac{1}{2^{n}} + \frac{1}{3^{n}} + \frac{1}{4^{n}} < \frac{1}{365} $$Find an N whereby all n \geq N give correct outcomes I thought that as long as ... 1answer 55 views ### Show that \lim \inf a_n\le\lim\inf s_n. Let \{a_n\} be a bounded sequence of real numbers. Let s_n=\dfrac{a_1+a_2+\cdots+a_n}{n}~\forall~n\in\mathbb N. Show that \lim \inf a_n\le\lim\inf s_n. The only definition I know of limit ... 0answers 18 views ### Convergence of the sequence v_n governed by$$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$$where ... 0answers 30 views ### Inequalities and sequences I ran across this problem in Gelfands Algebra today. 1/2 < 1/101 + 1/102 + ... + 1/200 < 1 He shows proof that this is true 1/2 < 1 - 1/2 + 1/3 - 1/4 + ... + 1/199 - 1/200 < 1 Again he ... 1answer 74 views ### the best constant in an inequality? I learnt how to show the below inequality by C-S inequality: k is from 0 to \infty If \sum a_{k}^{2}9^{k}\le 5 then \sum |a_{k}|2^{k}\le 3. next,I tried to show that 3 is the best possible ... 1answer 51 views ### How prove this \frac{1}{b_{k+1}b_{k+2}}+\frac{1}{b_{k+2}b_{k+3}}+\cdots+\frac{1}{b_{2k}b_{2k+1}}>\frac{1}{12345}, let sequence \{b_{n}\},and b_{n}>0,let$$S_{n}=b_{1}+b_{2}+\cdots+b_{n}\le n^{\frac{3}{2}},\forall n\ge 1$$show that ... 3answers 287 views ### Prove that \,\,\displaystyle\inf_{n\in\mathbb N}\sum_{k=0}^{p}\lvert\sin{(n+k)^p}\rvert>0 For any poistive integer number p, show that$$\inf\left\{ {\left\vert\sin{(n^p)}\right\vert+\left\vert\sin{(n+1)^p}\right\vert+\cdots+ \left\vert\,\sin{(n+p)^p}\right\vert\, :\,n\in ...
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 ...