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

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0answers
18 views

A property on finite sequences $1,1,x_2, x_3, \dots, x_{n-1}$ with $x_i \in \{0,1\}$

Consider a finite sequence $$x_0, x_1, \dots, x_{n-1}$$ with $n$ odd, all $x_i \in \{0,1\}$ and in particular $x_0 = x_1 = 1$. Furthermore, assume that the number of nonzero $x_i$ is even and $\leq ...
1
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1answer
44 views

Is $f(x)=\sum_{n=2}^{\infty} \frac{1}{n\ln(n)^x}$ continuous on $(1,\infty)$?

Is $f(x)=\sum_{n=2}^{\infty} \frac{1}{n\ln(n)^x}$ continuous on $(1,\infty)$? I have proven that the infinite series converges on $(1,\infty)$. I want to use the Weierstrass M-test to prove this ...
1
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0answers
21 views

Confusing multiplication of Fomal Power Series

I have the following product I want to multiply: ...
0
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1answer
23 views

Limit of a recursive sequence

Given $z_1>0$ and $a >0$ $z_{n+1} = (a+z_{n})^{1/2}$ To show $z_n$ is convergent and find its limit. The general approach would be to use PMI(Induction) to show that there exists an upper ...
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2answers
46 views

What is the sum in 37th bracket…?

I have a series like this $$\left(7^0\right)+\left(7+7^2+7^3\right)+\left(7^4+7^5+...+7^8\right)+\left(7^9+7^{10}+...+7^{15}\right)$$ I want to find the sum in the 37th bracket.Can anyone guys ...
2
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1answer
25 views

Show that, $\sum_{n=1}^{\infty}\left[\frac{\beta(2n)}{n}-\ln\left(\frac{n+1}{n}\right)\right] =…$

valid for all $s\ge 1$ $$\beta(s)=\sum_{n=0}^{\infty}\frac{1}{(2n+1)^s}$$ The particular value of $\Gamma\left(\frac{1}{4}\right)=3.6256099...$ Euler's constant is defined by $$\lim_{n \to ...
2
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2answers
41 views

Closed form for $\prod_{i=0}^{\infty}(1+x^{2^i})$

I've recently come across the infinite product $\prod_{i=0}^{\infty}(1+x^{2^i})$ and I was wondering if there is a closed form expression for this, or even if it diverges for all non-zero $x$. ...
3
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0answers
45 views

I discovered a sequence that should converge to $\pi$ , but how to prove that it really converges to $\pi$? [duplicate]

So yesterday I took paper and pencil and did some constructions of geometrical nature and I discovered a sequence that should converge to $\pi$. It goes like this: Define $a_1=\sqrt{2}$ and for ...
0
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0answers
9 views

Infinite Sum of Incomplete Gamma (with “z” varying)

I am trying to see whether the following series can be "reduced" or re-written in terms of well-known functions. This seems different from other questions on here in that the summation is going ...
1
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1answer
44 views

Integration of a polylogarithm: Is this function known?

I would like to integrate a polylogarithm of a given order $$\int dx \mbox{Li}_{n-1}(x)$$ suppose that the order is $n\le 0$ and $x\in(-\infty,0]$, so the function is bounded. I know that it can be ...
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1answer
51 views

Does a convergent sequence in theory ever reach its limit?

Completing a question on the sequence $\{a_n\} = \frac n{2n+1}$. Does $a_n$ in this sequence ever actually get to $\frac12$ officially?
0
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0answers
25 views

Proof of an integration equality and an infinite series equality derived thereof

I need hints on the proof of: $$\int_0^\infty\dfrac{\ln(x)^2}{1+x^2}{\rm{d}x}=\dfrac{\pi^3}{8}$$ and then: $$\sum\limits_{n=0}^\infty\left((-1)^n\dfrac{1}{(2n+1)^3}\right)=\dfrac{\pi^3}{32}$$ ...
0
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1answer
29 views

limit of $\sum_{j=n}^{4n}\binom{4n}j(1/4)^j(3/4)^{4n-j}$ as $n\to\infty$

I want to find out the value of the limit: $$\lim_{n \rightarrow \infty}\sum_{j=n}^{4n} \dbinom{4n}{j} \left(\frac{1}{4}\right)^j \left(\frac{3}{4}\right)^{4n-j}$$ I am not getting any hint: please ...
1
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1answer
60 views

Does the following series converges $\sum_{n=2}^{\infty} \frac{1}{n (\ln n)(\ln \ln n)^2} $?

Given series $$\sum_{n=2}^{\infty} \frac{1}{n (\ln n)(\ln \ln n)^2} .$$ To determine whether it is convergent or divergent. I tried with ratio test but it is inconclusive. Cauchy condensation test ...
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0answers
55 views

What is the flaw in the proof that the sum of all positive integers equals to $\frac{-1}{12}$? [duplicate]

According to this article: http://physicsbuzz.physicscentral.com/2014/01/redux-does-1234-112-absolutely-not.html ...the infinite series: $1 + 2 + 3 + 4 + \cdots$ does not actually equal ...
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5answers
212 views

What is the sum of all the Fibonacci numbers from 1 to infinity.

Today I believe I had found the sum of all the Fibonacci numbers are from $1$ to infinity, meaning I had found $F$ for the equation $F = 1+1+2+3+5+8+13+21+\cdots$ I believe the answer is $-3$, ...
0
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3answers
43 views

finding the sum of this series $(2n-1)^2(1/2)^n$

$\sum\limits_{n=1}^{\infty}(2n-1)^2(\frac{1}{2})^n$ I know via Wolfram that the sum is 17, but I'm not sure I've ever found the sum of such a series before. Any help is appreciated.
2
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1answer
54 views

Sequence ratio test.

The ratio test for sequences states that if $a_n$ is non-negative and $$\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = L$$ then $\lim a_n = 0$ if $L <1$ and $\lim a_n = \infty$ if $L >1$. The test ...
4
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1answer
64 views

Show that, $2\sum_{s=1}^{\infty}\frac{1-\beta(2s+1)}{2s+1}=\ln\left(\frac{\pi}{2}\right)-2+\frac{\pi}{2}$.

The Dirichlet beta function is defined as for Re(s)>0 $$\beta(s)=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)^s}.$$ Show that, ...
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0answers
17 views

State whether the following series converges or diverges

So for finding if the sequences converges or diverges I was going to use the limit comparison test. I was thinking with K/ (K$^2$)$^1/2$ but wouldn't that just be 1? Help please!
0
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3answers
48 views

Finding a Taylor Series Representation: $f(x)= \frac{x}{(1+4x^2)^2}$ [on hold]

Find Taylor series representations for the following function. For precisely what values of $x$ is the series representation valid? $$f(x) = \frac{x}{(1+4x^2)^2}$$
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0answers
17 views

Limit of Recursive sequences [on hold]

Given $Z_1 > 0$ and $a > 0$ $Z_{n+1} = (a+Z_n)^{1/2}$ To show $Z_n$ is convergent and find its limit. The general approach would be to use PMI(Induction) to show that there exists an upper ...
0
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1answer
19 views

Proving $\sum_{k=1}^{\infty}(-1) \ ^{ k+1} ak <\infty$ for $a_n\downarrow 0$.

Let $(a_n)$ be a sequence of positive real numbers such that for each $n\in\Bbb N$, $a_{n+1}\le a_n$, and $\lim\limits_{n\to\infty}a_n=0$. Prove that the following series is convergent: ...
-3
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1answer
59 views

Is the series $\sum_{n=1}^\infty\frac{n^n}{n!e^n}$ divergent? [on hold]

How I can show that the following series is divergent $$\sum_{n=1}^\infty\frac{n^n}{n!e^n}?$$ thank in advance.
7
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4answers
309 views

Ratio test for sequences, the other direction

Suppose I have a real sequence $x_n\to 0$. Is it true that: $$ \left|\frac{x_{n+1}}{x_n}\right|\to r<1 $$ for some $r\in\mathbb{R}$? If not, is it true that: $$\exists ...
3
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1answer
46 views

Fibonacci Pairs

Find all positive integer solutions to $y^2 - xy - x^2 = 1$ and $y^2 - xy - x^2 = -1$ I have written a C++ program to yield some solution for large constants. I must make conjectures based on the ...
3
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2answers
28 views

Convergence of the series $\sum (a^{1/n}-1)^{\lambda}.$

The series $$\sum_{n=1}^{\infty} (a^{1/n}-1)^{\lambda}$$ converges for $1.\lambda\geq0$ $2.\lambda\geq1$ $3.\lambda>1$ $4.\lambda\leq1$ I am confused about the convergence of the series. No ...
2
votes
2answers
49 views

conditionally convergent but not absolutely convergent series

I'm stuck on the following exercise: Let $\sum_{n=0}^{\infty} a_n$ be a series of real numbers which is conditionally convergent, but not absolutely convergent. Define the sets ...
0
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0answers
5 views

Integral operator with sub-exponential eigenvalue decay

I am looking for a positive-definite symmetric function $K: \Omega \times \Omega \to \mathbb{R}$ ($\Omega \subset \mathbb{R}$), such that $$ K(x,y) = \sum_{k=0}^\infty \exp(-\sqrt{k}) \phi_k(x) ...
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0answers
23 views

Reference for Coefficient Extraction of Multiple Sum

In a post here, the final answer is obtained by coefficent extraction of the quadruple sum. ...
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0answers
14 views

Convergence of a product of two infinite sums

Let $(\theta_j)_{j \in \mathbb{Z}}$ and $(\phi_j)_{j \in \mathbb{Z}}$ be two collections of real numbers indexed by integers. I am asked to show that $$\sum_{i \in \mathbb{Z}}\sum_{j \in ...
4
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1answer
31 views

Question on proving tight sequences.

I was just wondering how you would go about showing that a sequence of random variables is a tight sequence. For example suppose $X_{n}$ is distributed Exponentially($\lambda_n$) how would I show that ...
0
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1answer
20 views

If $\{u_n\} \to u$ and $\{v_n\} \to v$. Show that $\{\rho(u_n, v_n)\} \to \rho(u,v)$

Let $(X,\rho)$ to be a metric space in which $\{u_n\} \to u$ and $\{v_n\} \to v$. Show that $\{\rho(u_n, v_n)\} \to \rho(u,v)$ Proof: Suppose $\{u_n\} \to u$ and $\{v_n\} \to v$. This means that ...
4
votes
1answer
100 views

If $\sum a_n$ is convergent, is $\sum\frac{a_n}{n}$ absolutely convergent?

Assume that a series $\sum\limits_{n\geqslant1} a_n$ is convergent. Does this imply that $\sum\limits_{n\geqslant1}\frac{a_n}{n}$ is absolutely convergent? My first thought is no, but I'm having ...
3
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1answer
32 views

Definition of convergence of a sequence

Can this be a valid definition? For each $\epsilon>0$, there exists $K \in \Bbb N$ such that if $n\ge K$, then $|a_n-a|\le\epsilon$. Should I use "$<\epsilon$" or "$\le \epsilon$"?
3
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3answers
30 views

A different notion of convergence for this sequence?

I was thinking about sequences, and my mind came to one defined like this: -1, 1, -1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, 1, ... Where the first term is -1, and after the nth occurrence of -1 in the ...
4
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1answer
63 views

Prove that for $k>1$ $a_n$ is a perfect square

I'm having problems with this exercise. Let $k > 1$ be an integer. We define $(a_n)_{n \in \Bbb N_0}$ as: $$a_0 = 1$$ $$a_1 = 1$$ $$a_{n+2} = (k^2-2)a_{n+1}-a_n-2(k-2)$$ Prove that $\forall n \in ...
5
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1answer
42 views

Order of divergence of an infinite series

Let $(a_n)_{n\in\mathbb N}$ and $(b_n)_{n\in\mathbb N}$ be two real sequences such that $a_n\geq0$ and $b_n\geq0$ for all $n\in\mathbb N$; $\sum_{n=1}^{\infty} a_n=\infty$; $(b_n)_{n\in\mathbb N}$ ...
0
votes
1answer
17 views

Alternating sum of combinations of the n by consecutive k

Like in this question I have to prove that the alternating sum equals 0. If n is even the case is easy: there exist a bijection between the elements of the series and they cancel each other out. If n ...
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0answers
29 views

Convergence of $\sum \frac{a_n}{1+a_n}$ implies convergence of $\sum a_n$ for positive $a_n$. [duplicate]

I need to prove or disprove the statement. I think the statement is true. My attempt at a proof: From the definition of convergence: $$\forall \epsilon > 0 \quad \exists N \in \mathbb{N} \quad ...
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0answers
5 views

uper bound of limsup for one specific subsequence implies the sambe bound for continuous limsup

I was reading a paper about large deviations and they have some probability $p(\varepsilon)$ which depends on the parameter $\varepsilon >0 $ and they want to take limit when $\varepsilon \to 0$. ...
0
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0answers
16 views

Show that the absolute infinite series $\sum_{k=1}^{\infty} \lvert x_k \cdot y_k \rvert$

Show that the absolute infinite series $\sum_{k=1}^{\infty} \lvert x_k \cdot y_k \rvert$ converges if the absolute infinite series $\sum_{k=1}^{\infty} \lvert x_k \rvert$ converges too and the ...
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3answers
79 views

Does the infinite series $\sum_{k=1}^{\infty} \frac{1}{(4+(-1)^k)^k}$ converge?

I have been wondering if this infinite series converges $$\sum_{k=1}^{\infty} \frac{1}{(4+(-1)^k)^k}$$ I tried to put it in wolfram alpha but it says that the ratio test is inconclusive, but when I ...
0
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0answers
20 views

Reference for finite sum of $k^{\alpha}$

in this answer, we are given a great formula for $\sum\limits_{k=1}^n k^{\alpha}$ for all real alpha. For a paper I'm writing, I need a reference for a textbook or a paper which contains this result ...
0
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1answer
22 views

Should monotone convergence theorem say uniformly bounded?

saz pointed out to me the difference between bounded and uniformly bounded: $Y$ is uniformly bounded: there exists $C>0$ such that $|Y_n| \leq C$ for all $n \in \mathbb{N}$, i.e. $$|Y_n| ...
0
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2answers
35 views

Convergent of $\frac {\ln n} {n^p}$

Is the series $$\sum_{n=1}^{\infty}\frac {\ln n} {n^p},$$ $\ p \in \Bbb N$, convergent or divergent, and how to prove it?? I think I can use the integral test but maybe another test is easier than ...
1
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2answers
58 views

If $(a_n)$ is positive and $\sum\limits_n \frac{a_n}{1+a_n}$ converges then $\sum\limits_n a_n$ converges

I need to either prove the following or find a counterexample. I really hope you can help, I cannot figure it out. Let $(a_n)$ be a positive sequence. $$\sum_{n=1}^\infty \frac{a_n}{1+a_n}$$ ...
0
votes
0answers
49 views

What explains this repeating pattern in the difference between a Riemann zeta zero related sequence and its conjectured asymptotic?

As the starting point for my experiment I assumed that the imaginary parts of the Riemann zeta zeros are of the form: $$\Im \{ \rho_n \} = \frac{2\pi}{\log x_n}$$ where $x_n$ is unknown. Therefore I ...
0
votes
1answer
19 views

Question Concerning Fourier Series

I was following the derivation of the basic Fourier series using orthogonal function. For the set of orthogonal functions $\{\phi_n\}$, say the function $f$ can be defined as: $$f(x) = c_0 \phi_0(x) ...
2
votes
0answers
25 views

Show that $ 3P_{\lceil n \rceil}-2=\sum_{k=1}^{A}\left(4-\left\lceil \frac{\pi(k)}{n}\right\rceil \right)^2 $

We proposed a formula for calculating nth prime number using the prime counting function. Where $\lfloor x\rfloor$ is the floor function and $\lceil x\rceil$ is a ceiling function. $\pi(k)$ is prime ...