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

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3
votes
4answers
42 views

Most natural way to prove $\sum_{n=1}^{\infty}\frac{1}{n+2}$ diverges

I don't know how my teacher wants me to prove that $$\sum_{n=1}^{\infty}\frac{1}{n+2}$$ diverges. All I know is that I have to use the $a_n>b_n$ criteria and prove that $b_n$ diverges. I tried ...
6
votes
1answer
80 views

Prove that sum is convergent

How to prove that the following sum is convergent? $$\sum_1^\infty\frac{\sin(n + \ln{n})}{n}$$ I tried to use formula $$\sin(x+\ln{x}) = \sin{n}\cos{\ln{n}} + \sin{\ln{n}}\cos{n}$$ and $$\sum_1^N ...
6
votes
1answer
307 views

Elementary proof of prime number theorem?

From Wikipedia: "The prime number theorem is also equivalent to: $$lim_{x \rightarrow \infty} \frac{\psi(x)}{x}=1$$ where $$\psi(x) = \sum\limits_{n \leq x} \Lambda(n)$$ is the Chebyshev function. ...
0
votes
1answer
45 views

Prove $\frac{\gamma}{4}+\ln\left[\frac{\Gamma(1/4)}{4} \right]=\sum_{n=2}^{\infty}\frac{(-1)^n\zeta(n)}{2^{2n}n}$

The originate idea of this formula is from here (1) $$\frac{\gamma}{4}+\ln\left[\frac{\Gamma\left(\frac{1}{4}\right)}{4} \right]=\sum_{n=2}^{\infty}\frac{(-1)^n\zeta(n)}{2^{2n}n}$$ We arrived at ...
0
votes
0answers
13 views

check for relationship duplicate numbers

I 'm a programmer C# and look for one to formulate mathematical search numbers related to avoid duplicate relationships I need to make a relationship of users in the database and for that I would not ...
0
votes
2answers
33 views

Recursive sequence nth element formula

What is the $n$th element of this sequence: $$S_n = S_{n-1} + (c_1 - S_{n-1})c_2$$ where $c_1$ and $c_2$ are constants and $S_1=0$. Thank you,
2
votes
1answer
49 views
0
votes
1answer
27 views

Is the sequence of functions $g_n=ng_1(nx)$ a cauchy sequence?

Given a function $g_1(x) \in \mathcal L^2(\Bbb R)$ that satisfies: $$\int_{-\infty}^{\infty}dx \space g_1(x)=1$$ one can define a sequence of functions $g_n=ng_1(nx)$. Does $g_n(x)$ define a ...
1
vote
1answer
48 views

Show that the series converges uniformly

Show that $\sum_{n=1}^{\infty} \sin \left(\dfrac{x}{n}\right)^2$ converges uniformly on $[a,-a]$ , $a\in \mathbb{R}$. My attempt: Using Taylor's formula, we have:$$ \sin\left(\dfrac{x}{n}\right)^2 ...
1
vote
1answer
35 views

Prove/disprove converge series

Can you help me or give me a hint with this, I don't know from where to start: prove/disprove this: $$\sum_{n=0}^\infty \frac{1}{(2n)!}=\frac{e^1+e^{-1}}{2}$$ Thanks!
4
votes
2answers
45 views

Show that the sequence $x_n=\Big[\frac32x_{n+1}\Big]$, $x_1=2$ contains an infinite set of odd numbers and an infinite set of even numbers.

I am having a hard time proving this Let $x_n$ be a sequence such that $x_{n+1}=\Big[\frac32x_n\Big]$ for $n\gt1$, where $[x]$ denotes the nearest integer function and $x_1=2$. Show that ...
0
votes
3answers
40 views

Finding a formula for a kth element in a sequence

I've setup a recurrence relation as part of a numerical analysis problem, and found that $$x_{n+1} = \frac{x_n+1}{2}$$ The notes then say that for $x_0=0$, it is easy to show that $$x_k = 1 - ...
-1
votes
2answers
41 views

Asymptotics of $\sum_{n}e^{-n^{2}}$.

Define the function $S(N)$ as $$S(N)=\sum_{n=0}^{N}e^{-n^{2}}$$ I am interested in the asymptotic behavior of $S(N)$ for large $N$. It is clear by the ratio test that $\lim_{N\rightarrow\infty}S(N)$ ...
1
vote
2answers
50 views

Writing sequences using $\sum$ and $\prod$ symbols

Rewrite the following expressions using $\sum$ or $\prod$ a) $(x-1)(x-4)(x-9)(x-16)....(x-900)$ b) $1/(6^3) + 1/(9^4) + 1/(12^5) + 1/(15^6) +......+ 1/(33^{12})$ For part a) I noticed that ...
5
votes
0answers
56 views

Show there are only a finite number of integers with $\dfrac{\prod_{i=1}^n a_i-1}{\prod_{i=1}^n (a_i-1)} $ an integer

Show, for each $n$, there are only a finite number of integral $(a_i)_{i=1}^n$ such that $2\le a_i \le a_{i+1}$ and $\dfrac{\prod_{i=1}^n a_i-1}{\prod_{i=1}^n (a_i-1)} $ is an integer. My question is ...
1
vote
1answer
27 views

Finding the values of $z$ such that $\sum_{n=0}^{\infty}(1+\sin{n})^nz^n$ converges

I'm trying to apply the nth root test to $$\sum_{n=0}^{\infty}(1+\sin{n})^nz^n$$ Hence I use that $\hat{R}=\left (\limsup |a_n|^{\frac{1}{n}}\right )^{-1}$ and get $$\hat{R}=\left (\limsup (1+\sin{n}) ...
1
vote
1answer
34 views

Is $Ax = (\alpha_1 x_1, \alpha_2 x_2, \alpha_3 x_3, \dots, \alpha_k x_k, 0, 0, \dots)$ a compact operator?

Is the operator $A$ defined by $$Ax = (\alpha_1 x_1, \alpha_2 x_2, \alpha_3 x_3, \dots, \alpha_k x_k, 0, 0, \dots)$$ a compact operator? It only has finitely non-zero dimensions, so does this mean ...
0
votes
3answers
21 views

If $f_n:X\to\mathbb{R}$ converge uniformly a $f$, then there is $N\in\mathbb{N}$ such that $n>N\to f_n(X)\subseteq U$.

Let $X$ be a compact, $U$ open set and $f:X\to\mathbb{R}$ continuous such that $f(X)\subseteq U$. If $f_n:X\to\mathbb{R}$ converge uniformly a $f$, then there is $N\in\mathbb{N}$ such that $n>N$ ...
2
votes
1answer
62 views

Laurent series of $\frac{e^z}{z^2+1}$

I cant figure out the laurent series of the following function. Let $f(z)= \frac{e^z}{z^2+1} $ and $|z|\gt 1$ $$\frac{1}{z^2+1}=\sum_{n=0}^{\infty}(-1)^nz^{-2n-2}$$ and $$e^z = ...
0
votes
1answer
22 views

Unable to choose functions for evaluating a limit using the Squeeze Theorem

Evaluate $$\lim_{n\to \infty}\dfrac{1}{\sqrt{n^2}}+\dfrac{1}{\sqrt{n^2+1}}+...+\dfrac{1}{\sqrt{n^2+2n}}$$ $$$$ I'm supposed to solve this problem using the Squeeze Theorem. I had selected the ...
65
votes
10answers
12k views

Nice proofs of $\zeta(4) = \pi^4/90$?

I know some nice ways to prove that $\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6$. For example, see Robin Chapman's list or the answers to the question "Different methods to compute ...
2
votes
2answers
77 views

Complicated series converges to $\pi$.

How do I get this result? $$\frac {426880 \sqrt {10005}}{\large \sum_{k = 0}^{\infty}\frac {(6k)!(545140134k + 13591409)}{(k!)^3 (3k)! (-640320)^{3k}}} = \pi$$ It seems formidable. Context: I came ...
1
vote
3answers
153 views

How can I calculate OEIS A144311 efficiently?

I'm looking for a way to calculate OEIS A144311 efficiently. In one sense or another, this series considers the number between "relative" twin primes. What do I mean by this? Well, the number $77$ ...
2
votes
3answers
85 views

Series convergence $\sum_{n=1}^\infty\frac{\sin^4n}{\ln(n+1/n)}$

Series $A = \sum_{n=1}^\infty\frac{1}{\ln(n+1/n)}$ diverges by the comparison test (wolfram). I want to compare $\sum_{n=1}^\infty\frac{\sin^4n}{\ln(n+1/n)}$ with series $A$. How can I prove that ...
1
vote
2answers
83 views

Finding a closed form for $\sum_{k=1}^{\infty}(-1)^{n-1}\frac{\zeta(2n+1)}{2^{2n+1}}$

I've been playing with series involving odd values of the zeta function. Some time ago I found the following closed form $$ \sum_{k=1}^{\infty}\frac{\eta(2k+1)}{2^{2k+1}}=\frac{1}{2}-\ln(2) $$ and ...
-1
votes
0answers
26 views

Laurent Series, How it is done

Suppose that a series $$\sum_{n=-\infty}^{\infty}x[n]z^{-n}$$ converges to analytic function $X(z)$ in some annulus $R_1<|z|<R_2$. That sum $X(z)$ is called the z-transform of $x[n]$ $(n=0,\mp ...
0
votes
1answer
31 views

How can I prove that it is an Entire Function

Prove that if $$ f(z)=\left\{ \begin{array}{ll} \frac{\cos z}{z^2-(\pi /2)^2} & \hbox{when} \; z\neq \mp \pi/2\\ -\frac{1}{\pi}, & \hbox{when} \;z= \pi/2. \end{array} \right. $$ ...
0
votes
1answer
19 views

Taylor series integration

I am having trouble with the following question: Integrate the Taylor series $$e^{(-t^2)} = \sum^\infty_{n=0} \frac{(-t^2)^n}{n!}$$ term-by-term to obtain the Taylor series for erf (error function) ...
0
votes
1answer
20 views

Prove $-\frac{\ln(1-x^2)}{1-x^2}=H_1x^2+H_2x^4+H_3x^6+H_4x^8+\cdots$

$H_n$ is nth the harmonic numbers $x<1$ (1) $$-\frac{\ln(1-x^2)}{1-x^2}=H_1x^2+H_2x^4+H_3x^6+H_4x^8+\cdots$$ A different approach of representing $\ln(x)$ let expand out the series ...
1
vote
1answer
24 views

rewriting product of power series

According to $$\Lambda(\tau;q)=B_0(\tau)+\sum_{i=1}^\infty B_i (\tau) q^i$$ we define $$[\Lambda(\tau;q)-B_0(\tau)]^m=\bigg[ \sum_{i=1}^\infty B_i (\tau) q^i ...
4
votes
2answers
48 views

For which values of real $\alpha, \beta$ does $\sum_{n,m \ge 1} \frac{1}{n^{\alpha}+ m^{\beta}}$ converge?

I was wondering how does the series $$\sum_{n,m \ge 1} \frac{1}{n^{\alpha}+ m^{\beta}}$$ behave for real $\alpha, \beta > 0$. My approach: firstly I considered the case $\alpha = \beta > 2$. ...
4
votes
1answer
61 views

Verify the correctness of $\sum_{n=1}^{\infty}\left(\frac{1}{x^n}-\frac{1}{1+x^n}+\frac{1}{2+x^n}-\frac{1}{3+x^n}+\cdots\right)=\frac{\gamma}{x-1}$

$x\ge2$ $\gamma=0.57725166...$ (1) $$\sum_{n=1}^{\infty}\left(\frac{1}{x^n}-\frac{1}{1+x^n}+\frac{1}{2+x^n}-\frac{1}{3+x^n}+\cdots\right)=\frac{\gamma}{x-1}$$ Series (1) converges very slowly, we ...
1
vote
3answers
70 views

Limit of $\sum \limits_{k=1}^{n} \frac{2^kn+2n^2+k}{2^{k+1}n^2+2^kk}$ when $n\to\infty$

I have to show the convergence of the series $$\lim\limits_{n \to \infty}a_n=\sum \limits_{k=1}^{n} \frac{2^kn+2n^2+k}{2^{k+1}n^2+2^kk}.$$ I am quite sure that the limit is 1.5. I wanted to show this ...
1
vote
2answers
42 views

$(f_n) $ converge uniformly on $X$.

Let $(f_n)$ be a sequence function continuous $f_n:X\to \mathbb{R}$ that converge uniformly on $D\subseteq X$ dense set. Then $(f_n) $ converge uniformly on $X$. can someone help me with this. If X ...
1
vote
0answers
9 views

Maclaurin series and sereis converge [duplicate]

$f(x)$ is a continuos fuction in $[0,1]$ and differentiable twice on $0$. $U_n=(-1)^{n}f(\frac{1}{n})$ I need to prove that: $1.$ if $f(0)=0$ then $\sum_{n=1}^\infty U_n$ converge. $2.$ if ...
2
votes
0answers
54 views

$\zeta(1)=\frac12\ln(2)$? Did I do something wrong?

I attempted to calculate $\zeta(1)$ and I got $\frac12\ln(2)$. $$\zeta(1)=\lim_{\epsilon\to0}\frac{\zeta(1+\epsilon)+\zeta(1-\epsilon)}2$$ ...
1
vote
3answers
110 views

calculating $\lim_{n\to\infty} \frac{1+\frac{1}{2}+…+\frac{1}{n}}{n}$

How can I prove that $\lim_{n\to\infty} \frac{1+\frac{1}{2}+...+\frac{1}{n}}{n}=0$? I can't use $1+\frac{1}{2}+\cdots +\frac{1}{n}\approx \log n$ I've tried to use the following: $\lim_{n\to\infty} ...
-2
votes
0answers
53 views
1
vote
1answer
36 views

Approximate integral using Taylor Series

I have to approximate this integral with an error lesser than 0.1 using Taylor Series. This is the integral: $$\int_0^1 \arctan(\frac{1}{x^{10}}) dx$$ If I understood, I have to determinate the Taylor ...
1
vote
0answers
17 views

Order Size estimation of converging sum used for approximation of logarithm

I know it can be shown that $\log n=\sum_{i=1}^\infty \frac{(n-1)^i}{in^i}$ for $\forall n\in\Re$ where $n\ge1$ For given natural m, I tried to find the order size of k = f(m,n) in order for the ...
-2
votes
0answers
27 views

Summation of Series [on hold]

The question is that you have to find $\sum_{r=1}^n rx^{1/r}$ I tried various things such taking log of the general term , differentiating but couldn't reach anywhere .
2
votes
1answer
21 views

Convergence of a Fourier series to a point

Consider the function $f\left(x\right)=1+x$, $x \in \left[-\pi,\pi\right]$ I have calculated its Fourier series to be $$f\left(x\right)=1+2\sum^{\infty}_{n=0}\dfrac{\left(-1\right)^{n+1}}{n}\sin ...
0
votes
1answer
44 views

Intuition behind proving that $\lim_{n \to \infty} \frac{1}{b_n} = \frac{1}{b}$

Okay, so I'm having a bit of an issue understanding Rosenlicht's proof that $$\lim_{n \to \infty} \frac{1}{b_n} = \frac{1}{b}$$ I've worked out the proof and I have written it down myself, but ...
1
vote
0answers
32 views

Radius of convergence of the series $\sum\limits_{-\infty}^{\infty}(2^{-n}+4^{-n}) z^n$

I'm trying to find for what values of $z\in\mathbb{C}$ the series $$\sum_{n=-\infty}^{\infty}(2^{-n}+4^{-n})z^n$$ converges. My main methods are the nth root test and the ratio test. I believe it can ...
0
votes
1answer
24 views

Solving differential equations using series

Suppose $v \in \mathbb{R}$ and $$y\left(x\right)=x^v\left(1+\sum^{\infty}_{n=1}a_nx^n\right)$$ where the series converges for any $x \in \left(-r,r\right), r>0$ If $y\left(x\right)$ is a solution ...
0
votes
3answers
29 views

Proving a subsequence doesn't converge

When I want to prove that a sequence doesn't converge by showing that it's subsequence doesn't converge , can i use the limit comparison test? (Usually used for series) . for example - $$ \sum_{n ...
1
vote
0answers
21 views

Proof this limit superior is finite.

Let $\{ w_n \}$ be a sequence of non-negative numbers and put $M_n=\sum_{k=1}^n w_k^2 \xrightarrow{n\to\infty} \infty $. Proof that $$\limsup_{n\to\infty} \dfrac{\ln \ln \sqrt{M_n \ln \ln M_n} }{\ln ...
0
votes
0answers
27 views

When finding Laurent series when to use partial fractions?

When finding the Laurent series of $$f(z):=\frac{1}{z(z-1)(z-2)}$$ valid in the region $1<|z-2|<2$ for example do we just use partial fractions to break $f(z)$ up and the just find the Laurent ...
0
votes
1answer
34 views

Properties of Bessel Functions

I have shown that the Bessel function $$J_n\left(x\right)=\dfrac{x^n}{2^n}\sum^{\infty}_{n=0}\dfrac{\left(-1\right)^kx^{2k}}{2^{2k}k!\left(n+k\right)!}$$ satisfies the property ...
2
votes
1answer
35 views

Which exponents r>0 is the limit finite

I am trying to find values of $r>0$ such that $\lim\limits_{n\rightarrow \infty} \sum\limits_{k=1}^{n^2}\frac{n^{r-1}}{n^r+k^r}$ is finite. I have tried to use integral methods for this limit such ...