Articles of sequences and series

$(a_n)$ is a monotone increasing sequence of integers. Prove that: $\lim_{n\to\infty}(1+\frac{1}{a_n})^{a_n}=e$

I’m asked to prove the question above. I need to show that if $(a_n)$ is an increasing sequence of integers then: $\lim_{n\to\infty}\left(1+\frac{1}{a_n}\right)^{a_n}=e$ I was thinking of showing that $\lim_{n\to\infty}(a_n)=\infty$ and then, by definition, I can say that there exists a natural number $N$, such that for every $n>N, a_n>0$ and so $(a_n)^\infty_{n=N}$ is a sub-sequence […]

Determine if the sequence is convergent or divergent

(a). The first sequence is shown as: $$a_n = n\sin\left(\frac 1 n \right)$$ $$\lim_{n\to\infty}(a_n) = \lim_{n\to\infty} n\sin\left(\frac 1 n\right) = \lim_{n\to\infty} \frac{\sin\left(\frac 1 n \right)}{\frac 1 n} = \lim_{n\to0} \frac{\sin(n)} n = 1$$ Thus converges. Is this correct? (b). The second sequence is: $$a_n = \frac{(-1)^n n}{n+1}$$ How to do the limit of this sequence?

Summation of a series

I’m trying to solve a recurrence relation and came across this term $\sum_{i=0}^n i9^i$? I thought this was a geometric series, but I guess it’s not. Is it possible to solve this?

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) […]