Articles of sequences and series

Evaluate the sum $\sum^{\infty}_{n=1} \frac{n^2}{6^n}$

Evaluate the sum $\sum^{\infty}_{n=1} \frac{n^2}{6^n}$ My approach : $= \frac{1}{6}+\frac{2^2}{6^2}+\frac{3^2}{6^3} +\cdots \infty$ Now how to solve this I am not getting any clue on this please help thanks.

Infinite product of recursive sequence

Let $a_{n+1}=\sqrt {(a_n+a_{n-1})/2}$ and $a_0=a_1=2$, how to prove convergence of the product $a_0 a_1 a_2 a_3…a_\infty$, and possibly find its value?

Expected number of dice rolls of an unfair dice to roll every side equally many sides

I am having trouble with solving the following problem: The probability that a $d$-sided dice lands on its $k$th side is equal to $p_k$ for $k\in \{k\in\mathbb{N},k≤d\}$ and $p_1+p_2+p_3+…+p_d=1$. Roll this dice (at least once) until every side is rolled equally many times. Find a function $F(p_1,p_2…)$ which gives the expected number of rolls $n$ […]

General term of this interesting sequence

A sequence of positive integer is defined as follows The first term is $1$. The next two terms are the next two even numbers $2$, $4$. The next three terms are the next three odd numbers $5$, $7$, $9$. The next $n$ terms are the next $n$ even numbers if $n$ is even or the […]

Find the value of $\lim_{n\to \infty}\sum_{k=0}^n\frac{x^{2^k}}{1-x^{2^{k+1}}}$.

If $0 \lt x \lt 1$ and $$A_n=\frac{x}{1-x^2}+\frac{x^2}{1-x^4}+…..+\frac{x^{2^n}}{1-x^{2^{n+1}}}$$ then Find $\lim\limits_{n\to \infty}A_n$.

$\sum_1^n 2\sqrt{n} – \sqrt{n-1} – \sqrt{n+1} $ converge or not?

This question already has an answer here: Prove or disprove: $\sum a_n$ convergent, where $a_n=2\sqrt{n}-\sqrt{n-1}-\sqrt{n+1}$. 6 answers

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, […]