Articles of sequences and series

Probability that no two consecutive heads occur?

A fair coin is tossed $10$ times. What is the probability that no two consecutive tosses are heads? Possibilities are (dont mind the number of terms): $H TTTTTTH$, $HTHTHTHTHTHTHT$. But except for those, let $y(n)$ be the number of sequences that start with $T$ $T _$, there are two options, $T$ and $H$ so, $y(n) […]

A Recurrence Relation Involving a Square Root

Consider the recurrence relation: $a_{n+1} = \sqrt{a_n^2 -k},$ where $k>0$, $n\in\{0,1,n:a_n^2\geq k\}$, and $a_0>0$ is known. Is it possible to obtain an expression for $a_n$ in terms of $n$?

Limit of the sequence $\frac{1^k+2^k+…+n^k}{n^{k+1}}$

This question already has an answer here: Evaluate $\lim\limits_{n\to\infty}\frac{\sum_{k=1}^n k^m}{n^{m+1}}$ 4 answers

Smallest number $N$, such that $\sqrt{N}-\lfloor\sqrt{N}\rfloor$ has a given continued fraction sequence

How can I find the smallest positive integer $N$, such that the continued fraction of $\sqrt{N}-\lfloor\sqrt{N}\rfloor$ begins with a given finite sequence containing a zero followed by positive integers ? For example, the sequence $[0,1,2,3,4]$ is given. We have to find the smallest number $N$, such that $\sqrt{N}-\lfloor\sqrt{N}\rfloor$ begins with $[0,1,2,3,4]$. The number $\sqrt{388}-\lfloor\sqrt{388}\rfloor$ has […]

Do such sequences exist?

I wish to know if there are real sequences $(a_k)$, $(b_k)$ (and if there are, how to construct such sequences) such that: $b_k<0$ for each $k \in \mathbb{N}$ with $\lim\limits_{k \rightarrow \infty} b_k=-\infty,$ $$\sum_{k=1}^\infty |a_k| |b_k|^n< \infty, \space\forall n\in\mathbb{N}\cup\{0\}$$ $$\sum_{k=1}^\infty a_k b_k^n=1, \space \forall n\in\mathbb{N}\cup\{0\}$$

How do we know Taylor's Series works with complex numbers?

Euler famously used the Taylor’s Series of $\exp$: $$\exp (x)=\sum_{n=0}^\infty \frac{x^n}{n!}$$ and made the substitution $x=i\theta$ to find $$\exp(i\theta) = \cos (\theta)+i\sin (\theta)$$ How do we know that Taylor’s series even hold for complex numbers? How can we justify the substitution of a complex number into the series?

Define integral for $\gamma,\zeta(i) i\in\mathbb{N}$ and Stirling numbers of the first kind

Consider the integral $$\int\limits_0^{\infty}e^{-x}x^k\ln(x)^n\dfrac{dx}x$$ For $n=3$ we have $$(-\gamma^2-2\zeta(3)-3\zeta(2)\gamma)\genfrac{[}{]}{0pt}{}{k}{1}+3(\gamma^2+\zeta(2))\genfrac{[}{]}{0pt}{}{k}{2}-6\gamma\genfrac{[}{]}{0pt}{}{k}{3}+6\genfrac{[}{]}{0pt}{}{k}{4}$$ where $\genfrac{[}{]}{0pt}{}{n}{k}$ is Stirling number of the first kind. For different n there are simmilar formulas. In the general case $$\int\limits_0^{\infty}e^{-x}x^k\ln(x)^n\dfrac{dx}x\in\bigoplus_{j=1}^{n+1}\genfrac{[}{]}{0pt}{}{k}{j}\mathbb{Z}[\gamma,\zeta(i)]_{n+1-j}$$ where $$\mathbb{Z}[\gamma,\zeta(i)]=\bigoplus_{j=0}^{\infty}\mathbb{Z}[\gamma,\zeta(i)]_{j}$$ is graded ring. Does anyone know a simple proof? Are there any ideas on how to generalize on multiple zeta values?

Series – Apostol Calculus Vol I, Section 10.20 #24

I am having a lot of trouble with these series questions. Up until this point, I had relatively little trouble with all the questions in the book. These seem to require knowledge about approximations of functions and other external experience-based knowledge, which I just don’t have yet. Determine convergence or divergence of the given series. […]

Asymptotic behavior of Harmonic-like series $\sum_{n=1}^{k} \psi(n) \psi'(n) – (\ln k)^2/2$ as $k \to \infty$

I would like to obtain a closed form for the following limit: $$I_2=\lim_{k\to \infty} \left ( – (\ln k)^2/2 +\sum_{n=1}^{k} \psi(n) \, \psi'(n) \right)$$ Here $\psi(n)$ is digamma function. Using the method detailed in this answer, I was able to compute simpler, related series: $$\lim_{k\to \infty} \sum_{n=1}^{k} \left (\psi'(n) -1/n \right) =1 $$ $$\sum_{n=1}^{\infty} \psi”(n) […]

Find $\lim (a_{n+1}^\alpha-a_n^\alpha)$

Assume $\alpha \in (0,1)$, and $\{a_n\}$ is a strictly monotone increasing positive series. and $\{a_{n+1}-a_n\}$ is bounded. Find $$\lim_{n \rightarrow \infty}(a_{n+1}^\alpha – a_{n}^\alpha)$$. My idea is first proving for rational numbers , then use a rational sequences to approximate real numbers. But I can only prove for rational numbers. If $\alpha \in \Bbb{Q} \cap (0,1)$: […]