# Tagged Questions

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### How to prove that $1- \frac{x^2}{n} \leq (1+\frac{x}{n})^n\cdotp(1-\frac{x}{n})^n$

How would I prove this inequality (assuming its true, its from a textbook) $$1 - \frac{x^2}{n} \leq (1+\frac{x}{n})^n\cdotp(1+\frac{-x}{n})^n$$ if $n > |x|$, $x\in R$ and $n\in N$ I first ...
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### prove ${a_n} = \root n \of {n!}$ is monotonically increasing to $\infty$

Prove ${a_n} = \root n \of {n!}$ is monotonically increasing to $\infty$ I already showed that $a_n$ diverges to infinity like this: I used to the lemma which says that if ...
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### An inequality about a sequence

Let $(a_n)$ be a sequence such that $a_0=1 , a_1=2 , a_{n+1}=a_n+\dfrac {a_{n-1}}{1+ a_{n-1}^2} , \forall n \ge1$ , then is it true that $52 < a_{1371} < 65$ ? $EDIT:-$ I am posing another ...
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### Show that the following series is less than $4 \pi^2 / 3$.

Show: For any $k = 0,1,2,...$, $$\sum_{i=0}^{i=k} \frac{(k+1)^2}{(i+1)^2 (k-i+1)^2} \leq \frac{4 \pi^2}{3}.$$
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### How can I cleverly use the reverse triangle inequality in this case?

Say, we have the following recurrence relation: $$x_{k+2}=4x_{k+1}+x_k(-3-2h\lambda)$$ with $x_0$ given, $x_1=(1+h\lambda)x_0, h$ small (step size), and $\lambda<0$. Here is the context is case ...
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### How do I prove the following inequality $\sum_{k=n+1}^{\infty}\frac{1}{k^{2}\log k}\leq\frac{1}{n\log n}$?

I would appreciate some help proving the inequality $$\sum_{k=n+1}^{\infty}\frac{1}{k^{2}\log k}\leq\frac{1}{n\log n}.$$ Thanks in advance!
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### Proof of $\left|\sum_{k=1}^N\dfrac{k!}{N^k}-\sum_{k=1}^N\frac{1}{kN}\right|\le\dfrac{1}{4}$

Is it possible to prove the following inequality? $$\left|\sum_{k=1}^N\dfrac{k!}{N^k}-\sum_{k=1}^N\frac{1}{kN}\right|\le\dfrac{1}{4}$$ Thanks
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### On an estimate of sequences with weights

Does there exist a $C > 0$ such that $$\sum_{n \geq 1} a_n \leq C \left( \sum_{n \geq 1} 2^n a_n^2 \right)^{1/4} \left( \sum_{n \geq 1} 2^{-n} a_n^2 \right)^{1/4}$$ for all $a_n \geq 0$ with ...
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### Prove the inequality for all $N$

Show that the following inequality holds for all integers $N\geq 1$ $$\left|\sum_{n=1}^N\frac{1}{\sqrt{n}}-2\sqrt{N}-c_1\right|\leq\frac{c_2}{\sqrt{N}}$$ where $c_1,c_2$ are some constants. I have ...
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### $e^x>1+(1+x)\log(1+x)$,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?