Articles of sequences and series

Given $a_1,a_{100}, a_i=a_{i-1}a_{i+1}$, what's $a_1+a_2$?

I’ve been given the following puzzle Let $a_1, a_{100}$ be given real numbers. Let $a_i=a_{i-1}a_{i+1}$ for $2\leq i \leq 99$. Further suppose that the product of the first $50$ is $27$, and the product of all the $100$ numbers is also $27$. Find $a_1+a_2$. I tried the following, looking at the sequence for a moment […]

Taylor expansion of $\frac{1}{1+x^{2}}$ at $0.$

I am trying to find the Taylor expansion for the function $$f(x) = \frac{1}{1+x^{2}}$$ at $a=0.$ I have looked up the Taylor expansion and concluded that it would be sufficient to show that \begin{eqnarray} f^{(2k)}(0) & = & (-1)^{k}(2k)! \\ f^{(2k+1)}(0) & = & 0 \end{eqnarray} The problem I am having is that the derivatives […]

Show that $f(x)=\begin{cases}1&\text{if }x\in\mathbb{Q}\\0 &\text{if }x\in \mathbb{R}\setminus\mathbb{Q}\end{cases}$ is discontinuous everywhere

Can any one help me to answer this question: Show that $$f(x)=\begin{cases} 1 &\text{if }x\in \mathbb{Q} \\ 0 &\text{if }x\in \mathbb{R}\setminus\mathbb{Q} \end{cases}$$ is discontinuous everywhere. Notice: use this theorem Let $f:D\rightarrow \mathbb{R}$ and let $c\in D$. Then $f$ is continous at $c$ if and only if, whenever $X_n$ is a sequence in $D$ that converges […]

Estimate a sum of products

Let $0 < \alpha < 1$. Show that for $\lambda > 0$ big enough $$\sum_{n=1}^\infty \prod_{k=1}^n \frac{1}{\lambda k^{-\alpha} + 1} < \infty$$ I think $\lambda = 1$ is enough. You could then use the estimation \begin{align*} \prod_{k=1}^n \frac{1}{ k^{-\alpha} + 1} &\le \prod_{k=1}^n \frac{1}{ n^{-\alpha} + 1} \\ &= \left(\frac{1}{ n^{-\alpha} + 1}\right)^n \\ &= […]

Exact value of $\sum_{n=1}^\infty \frac{1}{n(n+k)(n+l)}$ for $k \in \Bbb{N}-\{0\}$ and $l \in \Bbb{N}-\{0,k\}$

The other day I was playing around with infinite sums and I found that for any fixed, positive integer $k$ the equality $$\sum_{n=1}^\infty\frac{1}{n(n+k)} = \frac{1}{k}\sum_{n=1}^k\frac{1}{n}$$ I got this result by using partial fraction decomposition and then cancelling out everything but a finite number of sums. This led me to seek a similar finite sum expression […]

Manipulating double summation

In a problem in my book, the following equality is there: $$\sum_{n=0}^\infty\Big( \sum_{k_i\ge 0, \sum_{i=1}^\infty ik_i=n}\frac{x^n}{\prod_{i=1}^\infty k_i!(i!)^{k_i}}\Big)=\sum_{k_i\ge 0}\Big(\frac{x^{\sum_{i=1}^\infty ik_i}}{\prod_{i=1}^\infty k_i!(i!)^{k_i}}\Big)$$ where the summation runs over all sequences $k_1,k_2\cdots$ of non-negative integers containing finitely many non-zero entries. It is not clear to me how to this equality is obtained. I did work out something like $\sum_{j=0}^\infty\sum_{k=0}^jf(j,k)=\sum_{k=0}^\infty\sum_{j=k}^\infty […]

How to calculate discrepancy of a sequence

For $d\geq1$ let $I^d=[0,1)^d$ denote the $d$-dimensional half-open unit cube and consider a finite sequence $x_1,\ldots,x_N\in{I}^d$. For a subset $J\subset{I}$, let $A(J,N)$ denote the number of elements of this sequence inside $J$, i.e. $$ A(J,N)=\left|\{x_1,\ldots,x_N\}\cap{J}\right|, $$ and let $V(J)$ denote the volume of $J$. The discrepenacy of the sequence $x_1,\ldots,x_N$ is defined as $$ D_N=\sup_{J}{\left|A(J,N)-V(J)\cdot{N}\right|}, […]

$\lim_{n \to \infty} \mid a_n + 3(\frac{n-2}{n})^n \mid^{\frac1n} = \frac35$. Then find $\lim_{n \to \infty} a_n$.

This question already has an answer here: What is the value of lim$_{n\to \infty} a_n$ if $\lim_{n \to \infty}\left|a_n+ 3\left(\frac{n-2}{n}\right)^n \right|^{\frac{1}{n}}=\frac{3}{5}$? 3 answers

How to prove a sequence does not converge?

I want to show that the sequence $$a_n=\frac{1}{n}+(-1)^n$$ does not converge to a limit. I know that if a sequence $\left( a_n\right)_{n \in \mathbb{N}}$ converges to a limit L, then $\forall\epsilon >0\;\;\exists N\in\Bbb N \;\text{such that}\;\forall n\geq N, \;\;|a_n-L|\leq\epsilon$, but I am not sure how to use this to prove a sequence that does not […]

A question concerning dot product of sequences with a specific asymptotic growth.

This question was posted/originated after a failure of a more generic attempt here: Let $\alpha_n$ be a sequence of positive Reals. It is known that $$\alpha_n \sim \log(n)$$ Let $\beta_n$ be another sequence of positive Reals such that, $$\sum_{k = 1}^n\beta_k \sim \log(n)$$ Can we say/prove that $$\frac{\sum_\limits{k = 1}^n\alpha_k\beta_k }{\sum_\limits{k = 1}^n\beta_k} \sim \frac{1}{2}\log(n)$$ […]