Articles of sequences and series

Let $a_{2n-1}=-1/\sqrt{n}$ for $n=1,2,\dots$ Show that $\prod (1+a_n)$ converges but that $\sum a_n$ diverges.

Let $a_{2n-1}=-1/\sqrt{n}$, $a_{2n}=1/\sqrt{n}+1/n$ for $n=1,2,\dots$ Show that $\prod (1+a_n)$ converges but that $\sum a_n$ diverges. What I have found so far is that $\prod_{k=2}^{2n} a_n$=$3(1-\frac{1}{2\sqrt{2}})\cdots (1-\frac{1}{n\sqrt{n}})$ and $\prod_{k=2}^{2n+1} a_n$=$3(1-\frac{1}{2\sqrt{2}})\cdots (1-\frac{1}{n\sqrt{n}})(1-\frac{1}{\sqrt{n+1}})$ I’m considering using the theorem that if each $a_n \ge 0$, then the product $\prod(1-a_n)$ converges if and only if, the series $\sum a_n$ converges. […]

Prove that s is finite and find an n so large that $S_n$ approximate s to three decimal places.

$$S=\sum_{k=1}^\infty\left(\frac{k}{k+1}\right)^{k^2};\hspace{10pt}S_n=\sum_{k=1}^n\left(\frac{k}{k+1}\right)^{k^2}$$ Let $S_n$ represent its partial sums and let $S$ represent its value. Prove that $S$ is finite and find an n so large that $S_n$ approximate $S$ to three decimal places. Solution: first of all, I think that we Will use L’hopital rule and then use root test while starting to solve this. But […]

Sum rule for Modified Bessel Function

Answer to this question will greatly be appreciated if somebody solve it. Mathematica does not give the answer $$\sum_{n=1}^\infty \frac{\frac{1}{2}I_{n-1}(x)+\frac{1}{2}I_{n+1}(x)-I_{n}(x)}{n^2}$$

Explicit formula for the series $ \sum_{k=1}^\infty \frac{x^k}{k!\cdot k} $

I was wondering if there is an explicit formulation for the series $$ \sum_{k=1}^\infty \frac{x^k}{k!\cdot k} $$ It is evident that the converges for any $x \in \mathbb{R}$. Any ideas on a formula?

Limit of a sequence that tends to $1/e$

Possible Duplicate: Showing that $\frac{\sqrt[n]{n!}}{n}$ $\rightarrow \frac{1}{e}$ Any suggestions to find the following limit: $$\displaystyle\lim_{n\to\infty}\left(\frac{n!}{n^n}\right)^{\frac{1}{n}}$$ with basic tools of the calculus!

Limit of Root: Shifted

It is a well known fact that: $$\lim_{N\to\infty}\sqrt[n]{n}=1$$ But what about the shifted one up: $$\lim_{N\to\infty}\sqrt[n+1]{n}=1$$ And what about the shifted one down: $$\lim_{N\to\infty}\sqrt[n]{n+1}=1$$

Is there a sequence of primes whose decimal representations are initial segments of each other?

I.e., is there a sequence of primes whose decimal expansions have the following form: $$a_1,\ a_1a_2,\ a_1a_2a_3,\ a_1a_2a_3a_4, \dots$$ What about with the order of the digits reversed, so each number’s decimal representation is a final segment of the next one’s? (Or any other interesting variation of restrictions?) What about in other bases? In binary, […]

Continuity of Fixed Point

For all $a \in \mathbb{R}$, let $f_a: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be continuous and contractive, that is, there exists $\epsilon \in (0,1)$ such that $\left\| f_a(x)-f_a(y) \right\| \leq (1-\epsilon) \left\| x-y\right\|$ for all $x,y \in \mathbb{R}^n$. Assume that for all $x \in \mathbb{R}^n$, the mapping $a \mapsto f_a(x)$ is continuous. Now let $x_0 \in \mathbb{R}^n$ be […]

Constructing a convergent subsequence

Let $a_{mn}$ be a double sequence in $[0,1]$. I would like to know whether I can do the following operation. I start with a sequence $a_{m1}$ and construct a convergent subsequence $a_{m'1}$. Then consider a sequence $a_{m'2}$ and construct a convergent subsequence $a_{m''2}$. Continuing this operation for all $n=1,2,…$ and renaming, construct a subsequence ${m^*}$. […]

proving montonity and convergence of sequence en = (1 + 1/n)^n

Prove the following. What would be the summation formula be for the first part?