Articles of sequences and series

Prove an algorithm for logarithmic mean $\lim_{n \to \infty} a_n=\lim_{n \to \infty} b_n=\frac{a_0-b_0}{\ln a_0-\ln b_0}$

Take: $$a_0=x,~~~~b_0=y$$ $$a_{n+1}=\frac{a_n+\sqrt{a_nb_n}}{2},~~~~b_{n+1}=\frac{b_n+\sqrt{a_nb_n}}{2}$$ Then we obtain as a limit the logarithmic mean of $x,y$: $$\lim_{n \to \infty} a_n=\lim_{n \to \infty} b_n=\frac{x-y}{\ln x-\ln y}$$ I don’t know how to prove this. But I do know that numerically it fits really well. In fact, the best approximation is obtained if we take geometric mean of $a_n,b_n$: $$x=5,~~~~y=3$$ […]

How do I prove that $\lim_{n\to+\infty}\frac{\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}}{\sqrt{n}}=?$

let sequence $\{a_{n}\}$ such $a_{1}=1$,and $$a_{n+1}a_{n}=n,n\ge 1$$ show that $$2\sqrt{n}-1\le\dfrac{1}{a_{1}}+\dfrac{1}{a_{2}}+\cdots+\dfrac{1}{a_{n}}<\dfrac{5}{2}\sqrt{n}-1$$ (2): I consider we can find this limit $$\lim_{n\to+\infty}\dfrac{\dfrac{1}{a_{1}}+\dfrac{1}{a_{2}}+\cdots+\dfrac{1}{a_{n}}}{\sqrt{n}}=?$$ My try:since $$a_{n+2}a_{n+1}-a_{n+1}a_{n}=n+1-n=1$$ so $$a_{n+2}=\dfrac{1}{a_{n+1}}+a_{n}$$ so $$\dfrac{1}{a_{n+1}}=a_{n+2}-a_{n}$$ so \begin{align*}\dfrac{1}{a_{1}}+\dfrac{1}{a_{2}}+\cdots+\dfrac{1}{a_{n}}&=a_{1}+(a_{3}-a_{1})+(a_{4}-a_{2})+(a_{5}-a_{3})+\cdots+(a_{n+1}-a_{n-1})\\ &=a_{1}+a_{n+1}+a_{n}-a_{1}-a_{2}\\ &=a_{n+1}+a_{n}-a_{2} \end{align*} since $$a_{1}=1,a_{1}a_{2}=1\Longrightarrow a_{2}=1$$ so $$a_{n+1}+a_{n}-a_{2}\ge2\sqrt{a_{n+1}a_{n}}-1\ge 2\sqrt{n}-1$$ so left hand inequality is prove it.Now consider Right hand inequality,we only prove this $$a_{n}+a_{n+1}<\dfrac{5}{2}\sqrt{n}$$ since $$a_{n}a_{n+1}=n\Longrightarrow a_{n}+\dfrac{n}{a_{n}}<\dfrac{5}{2}\sqrt{n}$$ […]

A closed form for $T_N = 1 + \sum\limits_{k=0}^{N-2}{(N-1-k)T_k}$?

I’ve narrowed down a problem I am working on to the following recurrence: $$\begin{align*} T_0 &= T_1 = 1\\ T_N &= 1 + \sum_{k=0}^{N-2}{(N-1-k)T_k} \end{align*}$$ I’m stuck on how to close it up, or at least make it linear or $O(n\log n)$. Any clues as to what technique I can use to make the sum […]

Is it possible to generate a unique real number for each fixed length sequence of real numbers?

Let A be the set of all sequences of real numbers of size $n$. Does there exist an injection from A to R? I know this is possible if we are only considering integers instead of real numbers; But I am not sure if it is possible if we consider real numbers instead. For integers, […]

Show that for any $w \in \mathbb{C}$ there exists a sequence $z_n$ s.t. $f(z_n) \rightarrow w$

I want to prove the following: Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be holomorphic and non-constant. Then for $w \in \mathbb{C}$ there exists a sequence $(z_n)_{n \in \mathbb{N}} \subset \mathbb{C}$ with $lim_{n\rightarrow w}f(z_n) = w$. Which theorem can I use here? I know that by Liouville $f$ must be unbounded but does that help me? Can […]

$\displaystyle\Big(1-\frac{t}{n}\Big)^n$ is strictly increasing for $n>N$ and $t>0$

Show that $\exists N\in\mathbb N$ such that, $\displaystyle\Big(1-\frac{t}{n}\Big)^n$ is strictly increasing for $n>N$ $(n\in\mathbb N, t>0)$ Bernoulli Inequality didn’t help me I did; $\displaystyle\frac{\Big(1-\frac{t}{n+1}\Big)^{n+1}}{\Big(1-\frac{t}{n}\Big)^n}=\Big(1+\frac{t}{(n+1)(n-t)}\Big)^n\Big(1-\frac{t}{n+1}\Big)$ $\displaystyle\Big(1+\frac{t}{(n+1)(n-t)}\Big)^n\ge\Big(1+\frac{nt}{(n+1)(n-t)}\Big)$$\quad$Bernoulli-Ineq. but $\displaystyle\Big(1+\frac{nt}{(n+1)(n-t)}\Big)\Big(1-\frac{t}{n+1}\Big)=1+\underbrace{\frac{nt}{(n+1)(n-t)}-\frac{t}{n+1}-\frac{nt^2}{(n+1)^2(n-t)}}_{\text{doesn’t seem to be positive}}$ So it didn’t work, do you have any ideas, thanks in advance.

Trigonometric identities using $\sin x$ and $\cos x$ definition as infinite series

Can someone show the way to proof that $$\cos(x+y) = \cos x\cdot\cos y – \sin x\cdot\sin y$$ and $$\cos^2x+\sin^2 x = 1$$ using the definition of $\sin x$ and $\cos x$ with infinite series. thanks…

Limit: $\lim_{n\to \infty} \frac{n^5}{3^n}$

I need help on a homework assignment. How to show that $\lim_{n\to\infty} \left(\dfrac{n^5}{3^n}\right) = 0$? We’ve been trying some things but we can’t seem to find the answer.

Calculate sum of an infinite series

I have been struggling with this functional series. $$\sum_{n=1}^{\infty}{(-1)^{n-1}n^2x^n}$$ I need to calulate the sum.Any tips would be appreciated.

Prove that every bounded sequence in $\Bbb{R}$ has a convergent subsequence

I am to prove that every bounded sequence in $\Bbb{R}$ has a convergent subsequence. I am stuck not knowing how and where to start.