For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

learn more… | top users | synonyms (5)

3
votes
4answers
67 views

Analyze if this sequence converges: $\sum_{n=0}^{\infty}\frac{n^{2}+1}{n!}$

Analyze if this sequence converges: $\sum_{n=0}^{\infty}\frac{n^{2}+1}{n!}$ I have used ratio test: $\lim_{n\rightarrow \infty}\left |\frac{a_{n+1}}{a_{n}} \right |< 1$ $\Rightarrow$ $\lim_{n\...
0
votes
0answers
19 views

Series Relation

$\{A_n\}$ is a sequence of 4th order tensors. $lim_{n\rightarrow\infty}A_n = O_4$, where $O_4$ is the null 4th order tensor. The series $\sum_{n=1}^{\infty}A_n$ converge to a known tensor $B$. I ...
1
vote
3answers
28 views

Summation of a series involving the MOD function

Evaluate the sum of: $$\sum_{n=0}^{\infty} (n\bmod 3)\cdot 2^{-n}$$ Any idea how can this sum be evaluated?
-3
votes
1answer
57 views

help in real analysis [on hold]

How can I use this definition to prove that $a^{\frac{1}{n}}$ converge to $1$? where a >0
0
votes
0answers
31 views

Approximation of series using integral

In notes of statistical physics I found the following approximation $$\sum\limits_{n=0}^{\infty}F\left(n+\frac{1}{2}\right)\approx \int_{0}^{\infty}F(x)dx+\frac{1}{24}F'(0)$$ for $F$ such that the ...
1
vote
3answers
55 views

Boundedness and convergence of $x_{n+1} = x_n ^2-x_n +1$

Suppose that $x_0 = \alpha \in \mathbb{R}$ and $x_{n+1} = x_n ^2-x_n +1$. I am asked to study the boundedness of $(x_n)$ and then asked if $(x_n)$ converges. How can I show that $(x_n)$ is bounded? ...
2
votes
1answer
53 views

Check whether my proof is correct or not.

The problem is : If the series $\sum a_n ^2$ and $\sum b_n ^2$ be both convergent, prove that the series $\sum a_n b_n$ is absolutely convergent. Using A.M. > G.M. we have $(a_n ^2 + b_n ^2)/2 \...
1
vote
2answers
51 views

Generating function for $\sum_{n=0}^{\infty} n^{p} x^n$

I am trying to derive the generating function for $H(x,p) = \sum_{n=0}^{\infty} n^p x^n$ I am trying to solve it with the following logic: (Edited now, trying a new framing) Base case: $$H(x,0) = \...
0
votes
2answers
21 views

Problem in finding an example related to infinite series.

The problem is : Give an example of a divergent series $\sum u_n$ such that $\sum u_{3n}$ is convergent. Please help me in finding this example.Thank you in advance.
0
votes
1answer
16 views

What is the area between two first order taylor series approximations as they become closer to eachother

Let's say that $y=\sin{x}$. Then the first order taylor series approximation about $c$ is $g(x)=\sin{(c)}+\cos{(c)}(x-c)$. Note that this is also equivalent to the line tangent to the curve $\sin{x}$ ...
0
votes
0answers
22 views

Formula to explain table data.

This is part of an excel spreadsheet and I would like to use a formula instead of a table to calculate steps and probabilities and so forth. It is my understanding that the table below represents the ...
3
votes
2answers
92 views

What is the limit of the sequence: $n$-th root of the $n$-th Fibonacci number?

My computer can not calculate numbers large enough for this. If you take the $n$-th Fibonacci number $F_n$ and raise it to the $1/n$-th term, where does the sequence $F_n^{1/n}$ tend to? Examples: ...
5
votes
1answer
54 views

Deriving sum of powers formula using generating functions

Just for fun I wanted to try to derive a formula for the sum of $p$-powers using generating functions, but without using any literature or websites for help. However I do need a small push or hint. ...
2
votes
5answers
81 views

Monotonicity of the sequence $(a_n)$, where $a_n=\left ( 1+\frac{1}{n} \right )^n$

Define $a_n=\left ( 1+\frac{1}{n} \right )^n$ for $n\geq 1$. I want to show that it is increasing. First, we have $$\frac{a_{n+1}}{a_n}=\left ( \frac{1+\frac{1}{n+1}}{1+\frac{1}{n}} \right )^n\left ( ...
0
votes
1answer
58 views

Having no derangements — any advantage?

Is there any problem-solving advantage when a sequence has no derangements? In an Erd\"os proof of Sylvester-Schur he identifies a few exceptions which I contend would not happen if his sequence had ...
2
votes
3answers
47 views

Proof related to Harmonic Progression

The question is as follows: Let $m_1<m_2<m_3<\cdots<m_k$ be postive integers such that $\frac{1}{m_1}$, $\frac{1}{m_2}$, $\frac{1}{m_3}$, $\cdots$, $\frac{1}{m_k}$ are in arithmetic ...
3
votes
2answers
33 views

Finding total elements in a series

I have a confusion. How many terms in the following series are needed to make a sun greater than 5/2? 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + ......... Is there any ...
7
votes
2answers
167 views

Value of this convergent series: $\frac{1}{3!}+\frac2{5!}+\frac3{7!}+\frac{4}{9!}+\cdots$

What is the value of- $$\frac{1}{3!}+\frac2{5!}+\frac3{7!}+\frac{4}{9!}+\cdots$$ I wrote it as general term $\sum\frac{n}{(2n+1)!}$. As the series converges it should be telescopic (my thought). But ...
0
votes
0answers
49 views

For what values of $k>1$ does $\sum_{n>1} 1/(n^k\sin(n))$ converge?

For what values of $k>1$ does $\sum_{n>1} 1/(n^k\sin(n))$ converge?
2
votes
2answers
60 views

Evaluating the sequence

I am currently working on the problem. Find $$\sum_{n=1}^{\infty} \frac{x_{n}}{n+2}$$ when $x_{n+2}=x_{n+1}-\frac{1}{2}x_{n}$ with $x_{0}=2$, $x_{1}=1$. I was able to find $x_{n}$ to be $x_{n}=(\...
4
votes
5answers
110 views

Compute $1 \cdot \frac {1}{2} + 2 \cdot \frac {1}{4} + 3 \cdot \frac {1}{8} + \cdots + n \cdot \frac {1}{2^n} + \cdots $ [duplicate]

I have tried to compute the first few terms to try to find a pattern but I got $$\frac{1}{2}+\frac{1}{2}+\frac{3}{8}+\frac{4}{16}+\frac{5}{32}+\frac{6}{64}$$ but I still don't see any obvious ...
1
vote
1answer
38 views

Doubly infinite matrices $A=(a_{i,j})_{i,j=\infty}^{\infty}$

Let $A=(a_{i,j})_{i,j=\infty}^{\infty}$, where $$ \|A\|:=\sum_{r=-\infty}^{\infty}\sup_{j}|a_{j,j+r}|<\infty. $$ I want to show that for all matrices $\|AB\|\leq\|A\|\|B\|$. I obverse that $$ (AB)...
0
votes
0answers
34 views

A series as a Riemann's sum

Let $\gamma$ and $\omega$ be two strictly positive real numbers. Let $t$ be in $[0,1]$. For any $\Delta>0$ let $k_t$ be the integer $$ k_t = \left\lfloor\frac{t}{\Delta}\right\rfloor, $$ so that $...
-1
votes
2answers
71 views

Reciprocal of the sum of powers of $1/x$ [duplicate]

Incidentally, I found $$\frac{1}{\sum_{n=1} \frac{1}{x^{n}}} = (x-1)$$ where $x\ge 2$. Please direct me to how others have developed the relationship. My computer cannot compute more than X = ...
0
votes
2answers
88 views

Why does $\sum_{n=1}^{\infty} \frac{2^n+1}{5^n+1}$ converge?

Why does $\sum_{n=1}^{\infty} \frac{2^n+1}{5^n+1}$ converge? I've tried by using the ratio test but I don't get so far, I'm a little lost with it. Any help will be really aprecciated.
0
votes
1answer
42 views

Order of summation for shifted exponential function

I want to represent the function: \begin{equation} f(x)=e^{-a(x-b)^{2}} \end{equation} where, $0<a<1$, $x\in\mathbb{R}$, and $b\in\mathbb{R}$. As a power series for an integral I am working ...
1
vote
0answers
31 views

Proving that this recursively defined sequence converges.

The sequence is defined as such, with $a_1=1$, $$ a_{n+1} = \begin{cases} a_n + 1/n, & \mbox{if } a_n^2 \leq 2 \\ a_n - 1/n, & \mbox{if } a_n^2 > 2 \\ \end{cases}. $$ In the book, P.M. ...
1
vote
1answer
24 views

Local normal convergence equivalent to compact normal convergence

Let $X$ be an open subset of $\mathbb{R}^m$ and let $f_n\colon X\to \mathbb{C}$ be complex-valued functions. Then one has the following two notions: $\textbf{1.}$ The series $\sum\limits_{n=0}^{\...
5
votes
4answers
105 views

How do we show that $\ln{2}-\gamma=\sum_{n=1}^{\infty}{\zeta(2n+1)\over 2^{2n}(2n+1)}?$

$$\ln{2}-\gamma=\sum_{n=1}^{\infty}{\zeta(2n+1)\over 2^{2n}(2n+1)}\tag1$$ Any hints?
3
votes
0answers
93 views

Limits of $a_n, b_n, c_n$

Three given positive $a_1, b_1, c_1$, such that $a_1+b_1+c_1=1, \forall\ n,$ $$a_{n+1}=a_n^2+2b_nc_n, b_{n+1}=b_n^2+2a_nc_n, c_{n+1}=c_n^2+2a_nb_n$$ Prove $\{a_n\},\{b_n\}$ and $ \{c_n\} $ are ...
0
votes
0answers
47 views

Simplify Series composed by Noncommutative Matrices

Problem I need to find a simpler formula for the following series: S = $\sum_{a=1}^{\infty} \frac{1}{a} \sum_{b=1}^{a} X^{b-1}MX^{a-b} = \sum_{a=1}^{\infty} \frac{1}{a} X^{a-1} \sum_{b=0}^{a-1} X^...
1
vote
0answers
30 views

Operator theory to study a difference equation

I'm not an expert in operator theory (so I'm going to be very informal sorry), but I would like to be given some advice about a problem I have. Let $f$ be a function defined in $C^{\infty}(\mathbb{R})$...
-1
votes
0answers
26 views

$(a_n)$ is bounded and $\forall n \in \Bbb N: a_n \neq 0\Rightarrow\frac{1}{(a_n)}$ is bounded [on hold]

I need to proof:$(a_n)$ is bounded and $\forall n \in \Bbb N: a_n \neq 0 \Rightarrow \frac{1}{(a_n)}$ is bounded
2
votes
1answer
30 views

Asymptotic solution of the equation $\gamma_{i+2} + 4\gamma_{i+1} + \gamma_{i} = \frac{Kh^2}{12}$

I'm struggling with the following equation, I'm interested in an asymptotic solution: $$\gamma_{i+2} + 4\gamma_{i+1} + \gamma_{i} = \frac{Kh^2}{12}$$ Where $K$ is known constant, when $h \rightarrow ...
4
votes
0answers
33 views

Division of a square and value of a disk

I cam across this problem and I really don't know how to solve it. So you start with a square that has value 1. You divide this square in 4 so that each new square has a new value, as given by the ...
-3
votes
0answers
48 views

$\sum_{m\in \mathbb{Z}} e^{-im^2 t} e^{i m z} =? $ [on hold]

Can anyone sum up this series? $f(z, t) = \sum_{m\in \mathbb{Z}} e^{-im^2 t} e^{i m z} . $
0
votes
5answers
62 views

Does this expression diverge or converge?

I have the following expression: $$\lim_{n \to \infty} \frac{2}{n^2} \ {\sum_{i=1}^{n}{\sqrt{n^2 - i^2}}} \ $$ I am not quite sure whether it will converge or diverge. Can somebody tell me how to ...
0
votes
2answers
73 views

Exact value of a series

Is it true that $$\sum_{k=1}^\infty \left(\frac 3 2\sqrt{k}-\sqrt{k+1/2}-\frac{1}{2}\sqrt{k-1}\right)=\frac{(\sqrt2-4) \zeta(3/2)+4 \pi\sqrt 2}{8 \pi}?$$ (this is in regards to the question: ...
1
vote
0answers
43 views

Limit at infinity of a function series

In my researches I got stuck on two similar calculations, and I'd like to deal with them in one fell swoop. 1. I want to say that $$ \lim_{x \to \infty} \sum_{n > 1} z_n \!\!\!\sum_{\substack{d \...
1
vote
3answers
83 views

Will the expression $\sum_{i=1}^{n}{\frac{i^{2}}{n^{2}}}$ converge as n approches infinity?

I have the following expression: $$\lim_{n \to\infty}\ \sum_{i=1}^{n}{(\frac{i}{n})^{2}}$$ I am not quite sure whether it will converge or diverge. Can somebody tell me how to figure it out?
0
votes
0answers
26 views

Generalization of Mill's theorem

Mill's theorem states that there exists a positive real number A such that the floor of the double exponential function $A^{3^n}$ are primes for all positive integers n. The value of A is ...
0
votes
0answers
54 views

Challenging Series example [duplicate]

Let $\{x_n\}$ be a decreasing sequence such that the series of $x_n$ converge. Show that the limit as $n$ approaches infinity of $\{nx_n\}$ equals zero.
-1
votes
2answers
129 views

how is it that $\int_0^x 2\pi r\ dr$ is equal to the area of a circle [on hold]

I'm studying calculus and I'm having some basic questions, this one is regarding the area of a circle. we know, from some guy, that the circumference of a circle is $2 \pi r$ and the area can be seen ...
0
votes
2answers
42 views

I need help understanding The Integral Test for series

For the following Series I have to show that the series qualifies for The Integral Test, then use it to determine if the series converges or diverges. here's my work where I apply the Integral Test, ...
0
votes
2answers
42 views

How do you work out the product of this sequence?

I got a question in my maths paper and I didn't know how to answer it. This was the question: What is: $(1+\frac{1}{2}) (1+\frac{1}{3}) (1+\frac{1}{4}) $... All the way up to 98 factors a) What is ...
0
votes
0answers
21 views

Where were my mistakes when I've combined $\eta(s)=\left(1-2^{1-s}\right)\zeta(s)$ and the Fourier series for the fractional part?

Let $s=x+it$ the complex variable, thus we are denoting $\Re s=x$. Combining the identity $$\eta(s)=\left(1-2^{1-s}\right)\zeta(s),$$ that holds for $0<x<1$, where $\zeta(s)$ is the Riemann Zeta ...
1
vote
0answers
68 views

Analytical expressions for extreme values of $f(x):=\log(2)\left(\sum_{k=-\infty}^\infty 2^{k+x}e^{-2^{k+x}}\right)-1$

The function $f(x):=\log(2)\left(\sum_{k=-\infty}^\infty 2^{k+x}e^{-2^{k+x}}\right)-1$ is a periodic function. Numerical optimization shows that the minimum and maximum of $f$ are approximately $-9....
0
votes
3answers
62 views

How to solve $x_{100}$ with $x_{n+1}=\frac{2x_n}{2+x_n}+1$ and $x_{1}=3$

How to solve $x_{100}$ with $x_{n+1}=\frac{2x_n}{2+x_n}+1$ and $x_{1}=3$? Can anybody shed light on this? regards.
2
votes
4answers
55 views

Proving convergence or divergence of series: Tips and Tricks

I currently write an article where I collect some tips for students for proving the convergence or divergence of series. What tips and tricks do you know or use or teach? Remark: I will add some ...
5
votes
4answers
142 views

A series with logarithms

Can we express in terms of known constants the sum: $$\mathcal{S}=\sum_{n=1}^{\infty} \frac{\log (n+1)-\log n}{n}$$ First of all it converges , but not matter what I try or whatever technic I am ...