Articles of sequences and series

derivation of fibonacci log(n) time sequence

I was trying to derive following equation to compute the nth fibonacci number in O(log(n)) time. F(2n) = (2*F(n-1) + F(n)) * F(n) which i found on wiki form the fibonacci matrix equation stated there but i stuck in deriving it. I understood the derivation till the (-1)^n = F(n+1)*F(n-1) – F(n)^2 but then how […]

Theorem 3.55 Rudin (rearrangement and convergence)

If $\sum a_n$ is a series of complex numbers which converges absolutely, then every rearrangement of $\sum a_n$ converges, and they all converge to the same sum. Proof: Let $\sum a_n’$ be a rearrangement , with partial sums $s_n’$. Given $\epsilon > 0$ there exist an integer $N$ such that $m \geq m \geq N$ […]

Some pecular fractional integrals/derivatives of the natural logarithm

(Be prepared for a very long post) I have deduced the following formula: $$D^{-n}\ln(x)=\frac{x^n(\ln(x)-n)}{(-n)!}=\frac{x^n(\ln(x)-n)}{\Gamma(-n+1)}$$ Where $$D^{-1}f(x)=\int f(x)dx$$ $$D^{-2}f(x)=\int\int f(x)dxdx$$ $etc$. $$D^0f(x)=f(x)$$ $$D^1f(x)=\frac d{dx}f(x)$$ $$D^nf(x)=\frac{d^n}{dx^n}f(x)$$ So the $n$th integral of $\ln(x)$ is given by my formula if $n$ is a natural number. Since the formula is continuous for $n\in\mathbb{R}$ when $n$ is not a negative integer, […]

Proof: A convergent Sequence is bounded

A part of the proof says that if $n\le N$, then, the sequence $x_n \le \max\{|x_1|,|x_2|,….,|x_{N-1}|\}$. I’m not capturing the intuition of the above. Even more perplexing is, if it the case where the sequence is monotonic decreasing, then for every $n\le N$, it is obvious that $x_n$ will not be $\le \max\{|x_1|,|x_2|,….,|x_{N-1}|\}$. I greatly […]

Inequality with two sequences

Let $x_1,\dots,x_n$ and $y_1,\dots,y_n$ be two increasing sequences of nonnegative real numbers with $x_i\leq y_i$ for all $i$. Is there a constant $c>0$ (independent of $n$) for which there exists some $r\geq 0$ (possibly dependent on $n$ and the sequences) such that $$\sum_{i: x_i\leq r\leq y_i}y_i+\sum_{i:x_i\geq r} x_i\geq c\sum_{i=1}^n y_i?$$ This is a discrete version […]

Test for the convergence of the sequence $S_n =\frac1n \left(1 + \frac{1}{2} + \frac{1}{3} + \cdots+ \frac{1}{n}\right)$

$$S_n =\frac1n \left(1 + \frac{1}{2} + \frac{1}{3} + \cdots+ \frac{1}{n}\right)$$ Show the convergence of $S_n$ (the method of difference more preferably) I just began treating sequences in school, and our teacher taught that monotone increasing sequence, bounded above and monotone decreasing sequences, bounded below converge. and so using that theorem here.. I found the $$(n+1)_{th} […]

Riesz's Lemma on $l_\infty$

Riesz’s Lemma says that for a Banach space $X$, a proper closed subspace $Y\subset X$, and $\epsilon\geq 0$ there exist $x\in X$ with $\|x\|=1$ and $d(x,Y)\ge 1-\epsilon$. I have to find a closed subset in $l_\infty$ that demonstrates that the above result is not true for $\epsilon=0$. That is, construct $Y$ such that for all […]

Asymptotic behavior of $\sum_{j=1}^n \cos^p(\pi u j)$ for large $n$ and $p$?

Consider the sum $$S=\sum_{j=1}^n \cos^p(\pi u j),$$ where $n$ and $p$ are positive integers and $u$ is irrational. Let’s say $p$ is even. I’m interested in the asymptotic behavior of this for $n$ and $p$ both large. This is my attempt to make a finite sum similar to the series in this problem that might […]

Find the limit of a recursive sequence

Let $(u_n)_n$ be a real sequence such that $$ u_{n+2}=\sqrt{u_{n+1}}+\sqrt{u_{n}},\,u_0>0,\,u_1>0. $$ Fisrt, it is easy to check that $(u_n)_n$ is well defined and $u_n>0$ for all $n\in\mathbb{N}$. The question now is show that $$ \exists p\in \mathbb{N}\,;\,\forall n\in\mathbb{N},\,n\geq p\implies u_n>1. $$ From this, we can deduce the limit of the sequence $(u_n)_n$.

Proof of Divergence for a Sequence

Previous related question. After I negated the definition of a convergent sequence, I ended up with the following mathematical statement: $$\exists\ \epsilon > 0,\ \forall\ N \in \mathbb R\ \exists\ \mathbb N \ni n > N : |x_n – l| \ge \epsilon$$ Is this correct? I’d like clarification… Anyway, I’m now asked to use my […]