Articles of sequences and series

Bounding sum of reciprocals of the square roots of the first N positive integers

I am trying to derive the following inequality: $$2\sqrt{N}-1<1+\sum_{k=1}^{N}\frac{1}{\sqrt{k}}<2\sqrt{N}+1,\; N>1.$$ I understand for $N\rightarrow \infty$ the summation term diverges (being a p-series with p=1/2), which is consistent with the lower bound in this inequality being an unbounded function in $N$. With respect to deriving this inequality, it is perhaps easier to rewrite it as $$2\sqrt{N}-2<\sum_{k=1}^{N}\frac{1}{\sqrt{k}}<2\sqrt{N},\; […]

Fibonacci and Lucas series technique

Well, I have the following two problems involving Fibonacci sequences and Lucas numbers. I know that they share the same technique, but I don’t have clear the procedure: $$f_n = f_{n-1} + f_{n-2}: f_0 =0, f_1=1$$ $$l_n=l_{n-1} +l_{n-2}:l_0=2,l_1=1$$ Now, I want to prove that: $$\sum\limits_{k=0}^nf_k= f_{n+2}-1 $$ $$\sum\limits_{k=0}^n l_k^2= l_nl_{n+1} +2$$ My question is, what […]

Sum of Gaussian Sequence

I am looking for a closed form or an estimation for the sum of a Gaussian sequence expressed as $$ \sum_{x=0}^{N-1} e^{\frac{-a}{N^2} \: x^2} $$ where $a$ is a constant positive integer. The interesting part is that I have simulated this sequence using MATLAB for $N=0$ to $10^6$ and found that the result is linear […]

How to evaluate this limit? Riemann Integral

$$\large f(x)= \lim_{n\rightarrow \infty}\left( \dfrac{n^n(x+n)\left( x+\dfrac{n}{2}\right)\left( x+\dfrac{n}{3}\right)… \left( x+\dfrac{n}{n}\right)}{n!(x^2+n^2)\left( x^2+\dfrac{n^2}{4}\right)\left( x^2+\dfrac{n^2}{9}\right)…\left( x^2+\dfrac{n^2}{n^2}\right)}\right)$$ $x\in R^+$ Find the coordinates of the maxima of $f(x)$. My Work: Is the method correct? Is there an easier way?

Sum $\sum_{n>1} g(n)/n$ over a set of with density $>1/2$

If $S=\{x_i\}$ is a set of positive integers with asymptotic density in the positive integers strictly greater than $1/2$, and $$ g(n)=\begin{cases}-1&\text{if }\quad n\in S\\ 1&\text{otherwise} \end{cases} $$ Must the series $$ \sum_{n>1} g(n)/n $$ diverge?

Proof Check: Every Cauchy Sequence is Bounded

Sorry if I keep asking for proof checks. I’ll try to keep it to a minimum after this. I know this has a well-known proof. I understand that proof as well but I thought I’d do a proof that made sense to me and seemed, in some ways, simpler. Trouble is I’m not sure if […]

characterisation of compactness in the space of all convergent sequences

I go through a proof of the following. Let $(\ell_1,d)$ be the metric space of all sequences $x = (\xi_i)_{i \in \mathbb{N}}$ with $\sum_{i=1}^{\infty} |\xi_i| < \infty$ and the metric $$ d(x,y) = \sum_{i=1}^{\infty} |\xi_i – \eta_i|, \qquad x = (\xi_i), y = (\eta_i). $$ Theorem: A subset $M$ of $l_1$ is totally bounded (pre-compact) […]

Divergence of Reciprocal of Sequence knowing the Asymptotic Density

In a paper, it is stated that if a particular sequence of positive integers $a_1<a_2<\dots$ satisfies that the number of $a_i$ which do not exceed $n$ is $o(\frac{n}{\log ^2(n)})$, then $\sum \frac{1}{a_i}$ converges. I cannot see why this would be true- it’s probably really simple but it’s evading me right now. Help would be appreciated.

Is $\sum_{k=0}^{r}\binom{r}{k}(r-k-1)^{r-k}(k-1)^{k-1}+(r-2)^r=0$ right?

How to prove $$\sum_{k=0}^{r}\binom{r}{k}(r-k-1)^{r-k}(k-1)^{k-1}+(r-2)^r=0$$ I met this function when I tried to give another proof of the known lower bound of Tur\’an functions of complete hypergraphs ( Based on a same construction, instead of using shifting method, I tried to count edges directly ) Here is the question: Define $a_0=-1$ and $a_1=1$. For all $r\geqslant2$, […]

How do we find the $n+2$th term of the series $1 + (3+5) + (7+9+11)+\dots$?

We have the series $1 + (3+5) + (7+9+11)+\dots$. We need to find the $n+2$th term and hence summation of the series up to this term. However hard we try we do not seem to be able to fit this series to a pattern.