Questions on whether certain series diverge, and how to deal with divergent series using summation methods such as Ramanujan summation and others.

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3
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1answer
33 views

Series and comparison test

If $a_n>0$ and $\sum a_n$ diverges, what can be said about $\displaystyle \sum \frac{a_n}{1+na_n}$? I cannot prove that it is convergent or divergent. I think it is convergent for some examples ...
0
votes
1answer
37 views

Test whether $\sum_{n=1}^{\infty}\frac{\ln{n}}{n}$ converges or diverges

I am trying to solve this using an integral test, but I am unsure whether or not this is correct. Let $f:[2,\infty)\to\mathbb{R}$ be defined by $f(t)=\frac{\ln{(t)}}{t} >0\ \forall t\geq2$. Now ...
2
votes
1answer
35 views

$\sum_{n=0}^{\infty}n$ is divergence or convergence? [on hold]

Under cardinal arithmetic, the series $\sum_{n=0}^{\infty}n$ is equal to $\aleph_0$. Wouldn't this contradict with the series is divergence in calculus?
1
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3answers
40 views

Is sequence limited and what is limit

I am stuck at one problem. So I have to check if sequence is convergent. $$\frac{2^x}{x!}$$ My thinking was to calculate limit and if limit exists it's convergent, but I am struggling with this: ...
5
votes
3answers
108 views

Is there any method to get a finite sum for $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots$?

As we can see on Wikipedia, there are some algebraic methods that give us finite sums for the Grandi's series $$1-1+1-1+1-1+1-1+\cdots$$ Let $S$ be the sum of the Grandi's series. Then ...
0
votes
0answers
25 views

Divergence of series

Given that a is a non 0 real number and ck is a series of real number for all k natural number where $\sum_{k= 1}^{\infty} c_k^2 = \infty $. If $c_k \to 0$, proof that $lim _{n \to ...
1
vote
2answers
42 views

Determine whether or not $\sum_{k=1}^{\infty} \frac{1}{k- \mathrm{e}^{-k}}$ converges.

I have the following so far Let $a_k = \frac{1}{k- e^{-k}}$. Now, $\lim_{k \to \infty}a_k=0 \implies$ $\sum a_k$ can either converge or diverege. We must thus do further tests to determine whether ...
2
votes
1answer
21 views

Series of Sequence which always diverges

Suppose {$a_n$} is a sequence with $a_n>0$. For each $k$ in $\Bbb{N}$, set $$b_k = \frac{1}{k} \sum_{n=1}^{k}a_n$$ then woud $\sum_{k=1}^{\infty}b_k$ always diverge? I want to use Converge ...
-2
votes
0answers
63 views

What do you think of my proof?

I wrote this proof when I was still in high school (I just graduated). Never had the chance to have it checked by anyone. Therefore, I posted it here! :) Please edit if necessary! Our goal is to ...
1
vote
1answer
47 views

Sum series of normal pdf's evaluated in normal inverse cdf's

Any hint about how does the following sum grow for k going to infinity? $\sum_{i=1}^{k-1} \phi[\Phi^{-1}(i/k)]$ I "feel" it grows as $k/\sqrt{4\pi}$... but I am not able to prove it. I have also ...
1
vote
2answers
61 views

written one way, the limit doesn't exist. written another, the limit equals 0?

Hi I'd greatly appreciate some help/clarification with this problem, I'm also open to suggestions about how to be more articulate when writing problems out in general. On my way to saying that the ...
1
vote
0answers
35 views

Representing $i$ with divergent series

Train of thought In computer science, for representing signed integers, we can use the two's complement representation, so that $11111111_{(2)} = -1$ over 8 bits with $x_{(2)}$ denoting binary ...
2
votes
3answers
49 views

A Series Fails The Test For Divergence, but is Still Divergent?

I know $ \sum_{n=2}^\infty \frac{1}{n*ln(n)} $ is divergent by the integral test or comparison test; however, I notice that it fails the Series Test For Divergence ($\lim_{n\to\infty}a_n \neq 0 ...
3
votes
1answer
127 views

Is the series $\sum \limits_{n=1}^{\infty} \sin(n^2)$ convergent?

Does the series $\sum \limits_{n=1}^{\infty} \sin(n^2)$ converge? I think I've tried everything, I have no more ideas.
0
votes
0answers
12 views

Mesh reconstruction in hole filling using gradient-based editing

I am being involved in a project filling hole in triangle mesh, which based on the paper "a robust hole-filling algorithm for triangular mesh".Unfortunately, I have been stuck with a problem for a few ...
3
votes
2answers
56 views

How to show $\sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} = 4 \int_{0}^{\frac{1}{2}} \frac{\arcsin^{2}(x)}{x} \ dx$?

$$\sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} = 4 \int_{0}^{\frac{1}{2}} \frac{\arcsin^{2}(x)}{x} \ dx.$$ Someone please show that this equation is correct !?
0
votes
1answer
54 views

Is it possible to determine whether this series is convergent?

One often comes across stability regions when looking at explicit and implicit Euler's method (for $\dot{x}=\lambda x$). But I have never come across such region for Verlet, say for the ODE ...
5
votes
4answers
86 views

Proving that the series $\sum_{k=2}^\infty \frac{1}{k \ln k}$ diverges?

I don't know how to show this. The terms go to zero, and I can't really show that the terms dominate $\frac{1}{k}$ (a series with these terms diverges). Any other ideas?
1
vote
3answers
51 views

Prove $\sum_{n=1}^{\infty}|a_{n}b_{n}|$ converges if $\sum_{n=1}^{\infty}a_{n}^{2}$ and $\sum_{n=1}^{\infty}b_{n}^{2}$ converge

This is a homework problem for an undergrad topology course. Let $l^{2}$ be the set of all real-valued sequences $(c_{n})$ where $\sum_{n=1}^{\infty}c_{n}^{2}$ converges. Let $(a_{n}),(b_{n})\in ...
0
votes
0answers
22 views

Expand function using Maclaurin's series(infinite form)

Expand the function f(x)=log(1+x) in powers of x in an infinite series stating the validity of such expansion for x belonging to (-1,1]. The question actually asks to show that cauchy's remainder or ...
0
votes
3answers
63 views

Does this sequence diverge to ∞?

The sequence $(a_n)_{n \geq 1}$ is defined as follows: $$a_n:= \begin{cases} 0 \quad \text{if} \quad n \quad \text{is odd}\\ n \quad \text{if} \quad n \quad \text{is even}\end{cases} \quad .$$ Does ...
3
votes
1answer
58 views

Does $\sum_1^\infty \frac{n}{n^2 + 4}$ converge or diverge? Is my solution correct?

Does $\sum_1^\infty \frac{n}{n^2 + 4}$ converge or diverge? I am confused because my friend insists the series converges conditionally. I think the series diverges. Here is my process and solution: ...
0
votes
1answer
35 views

Difficulty understanding Divergence Test

I'm studying Series and Diverge Test. But I'm having a problem understanding it. It says that, when the limit of it's partial sums is not equal to zero then it diverges. But then, there's also an ...
0
votes
3answers
55 views

Does the series $\sum (1+n^2)^{-1/4}$ converge or diverge?

The integral is $\int\left(\,1 + n^{2}\,\right)^{-1/4}\,{\rm d}n$ is not quite possible, so I should make a comparison test. What is your suggestion? EDIT: And what about the series $$ \sum\left(\, 1 ...
0
votes
4answers
63 views

Does the series $\sum \frac{1}{n\ (\ln(n))^{3/2}}$ converge or diverge?

Consider $$\sum \frac{1}{n\ \ln^{3/2}(n)}$$ The ratio test is inconclusive. The root test is inconclusive. And it seems right that $\frac{1}{n\ (\ln(n))^{3/2}}\leq\frac{1}{n}$ which diverges, but ...
28
votes
3answers
2k views

Is $\sum_{n=1}^\infty \frac{\sin(2n)}{1+\cos^4(n)}$ convergent?

As I was just checking this 'child prodigy' out on Youtube, I stumbled upon this video, in which Glenn Beck asks the kid to do the following proof: Further on, the kid starts sketching a proof ...
0
votes
2answers
97 views

Does the equality $1+2+3+… = -\frac{1}{12}$ lead to a contradiction? [duplicate]

Is $1+2+3+4+5.... = -\frac{1}{12}$ self-contradictory ? I've heared much that $1+2+3+.... = -\frac{1}{12}$, although the fact that this series is diverging. I saw a proof of it by a physicist. In ...
1
vote
1answer
69 views

Divergence of $u_{n+1}=1+\frac{n}{u_n}$

Let $u_n$ be defined by $u_0=1$ and $u_{n+1}=1+\frac{n}{u_n}$. It can be shown easily that if it has a limit, then it must be $+\infty$. Does $u_n$ diverge to $+\infty$ ? What I have tried : Let ...
2
votes
0answers
29 views

How to find the analytic continuation of this series?

I have the following series: $$ \sum_{n = 0}^{+\infty} \frac{n^2}{(n^2 + a^2)^{\epsilon}} $$ with $a\in \mathbb{R}$. How can I find its analytic continuation for $\epsilon \in \mathbb{C}$? In ...
1
vote
1answer
43 views

Examples of divergent series summed by means of the analytic continuation of the corresponding

For my Bachelor's thesis, I am investigating divergent series. This is (yet another) question on this topic. Apparently, a divergent series $$ S = \sum_{n=1}^{\infty} a_{n} $$ can be summed by means ...
1
vote
1answer
27 views

How to test convergence for a tetration series slightly below the harmonic series?

I have the following series to test convergence \begin{align} S_{\infty}= \sum_{n=1}^{\infty} \dfrac{1}{n} \left( \dfrac{1}{n} \right)^{ \left( \dfrac{1}{n} \right) } < \sum_{n=1}^{\infty} ...
0
votes
1answer
84 views

Can a finite value for $\int_1^\infty \exp(x^2)\,dx$ be defined?

Why should $$\int_1^{\infty}\exp(ix^2)dx,\int_1^{\infty}\exp(-ix^2)dx,\int_1^{\infty}\exp(-x^2)dx$$ converges but not: $$\int_1^{\infty}\exp(x^2)dx$$ Is there any way that assigns a value to ...
0
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0answers
47 views

Why does Cauchy's Root Test for convergence of infinite series require $\limsup$?

I'm confused about the reasoning behind Cauchy's root test for convergence of infinite series. It states that for any series $\{a_n\}$, if $C = \limsup_{n\rightarrow\infty}{\sqrt[n]{|a_n|}} < 1$, ...
1
vote
1answer
27 views

In an irreducible, aperiodic, null-recurrent Markov chain holds $\sum_n p_{ij}^{(n)} = \infty$

My lecture notes state the following theorem: Theorem 2. Let $(X_n)$ be an irreducible, aperiodic, null-recurrent Markov chain. Then $$\forall i,j \in S : p_{ij}^{(n)}\to 0 \text{ and } \sum_n ...
2
votes
1answer
30 views

Convergence study of a series of functions

I am studying the convergence of the series $$ \sum_{n=0}^{\infty}\frac{\sin (x^n)}{(1+x)^n} $$ where $x \in \mathbb R$. My initial approach was to use the ratio test, but I am not getting to ...
4
votes
1answer
73 views

Is it possible to sum the divergent series with prime coefficients?

This is a follow-up of this question. It is known that the divergent series $$ P := \sum_{n=1}^\infty p_n \qquad \text{where } p_n \text{ is the $n$th prime} $$ cannot be summed by means of (prime) ...
0
votes
2answers
44 views

Determining if series converges or diverges

The Series is For this series the ratio test is inconclusive. I have rewritten the series as Currently i am approaching the problem using limit test. I couldn't progress from this point. Any ...
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votes
0answers
8 views

Upper and lower Abel sums for bounded sequences

Let $a_k,k\geq 0$, be a bounded sequence and consider the ``upper and lower Abel sums": $UA(a):=\limsup_{x \to 1-}\sum_{k=0}a_kx^k$ and $LA(a)=\liminf_{x \to 1-}\sum_{k=0}a_kx^k$. Is it true that ...
4
votes
1answer
83 views

Short form of few series

Is there a short form for summation of following series? $$\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n\dfrac{(-1)^kn!\alpha^{2n}((2y-1)^{2k+1}+1)}{2^{2n+1}(2n)!k!(n-k)!(2k+1)}$$ ...
8
votes
1answer
88 views

If the generating function summation and zeta regularized sum of a divergent exist, do they always coincide?

One could assign a value to divergent series by means of several summation methods. One summation method we could consider is the generating function method. Let's sum, for example, the fibonacci ...
0
votes
3answers
73 views

Convergence of the series $\sum n!/(n^2+3)$

How can we test if this series diverges/converges? $$\sum_{n=1}^\infty\frac{n!}{n^2+3}$$ I tried D'Alembert's principle and tried to do $\frac{a_{n+1}}{a_n}$ but I'm stuck. Any help?
3
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0answers
171 views

Proof on why $0-1+2-3+4-\ldots\neq-1/4$

When reviewing my notes on series' convergence, I thought of applying a workaround on why $\sum_{n=0}^{\infty}(-1)^nn$ should or shouldn't be $-1/4$ (I recalled this page). I started by considering ...
0
votes
1answer
79 views

Convergence of the series $\sum_{n=1}^{\infty}\frac{1+n!}{(1+n)!}$ [closed]

I need to find out whether this series converges or diverges. $\sum_{n=1}^{\infty}\frac{1+n!}{(1+n)!}$ Can someone help how to solve it?
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2answers
34 views

Convergence test of certain series

I need to find out whether this sequence converges or diverges using limit comparison test. $\sum_{n=2}^{\infty}\frac{\sqrt{n+2}-\sqrt{n-2}}{\sqrt n}$ I've tried it with the use of sequence ...
3
votes
2answers
113 views

If a recursive sequence converges, must its inverse be divergent?

Suppose I have a recursive sequence $\displaystyle a_{n+1} = \frac{a_{n}}{2}$. Clearly, the sequence converges towards zero. Now, suppose I define an "inverse" sequence $\displaystyle b_{n+1} = ...
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votes
0answers
56 views

Strange sum divergent

Could you Find /Check the strange sum i have calculate in a unformal way it is like a analitycal continuation $$\sum _{k=1}^{\infty } (-1)^k k \log \left(\frac{k+3}{k+2}\right)=\frac{1}{6} (-36 \log ...
1
vote
3answers
91 views

Does $x^2+ x^4/(3\cdot4) + x^6/(3\cdot4\cdot5\cdot6) + \cdots$ have any compact form?

Is there any compact form for the following series $$F_1(x) = x^2+ \frac{x^4}{3\cdot4} + \frac{x^6}{3\cdot4\cdot5\cdot6} + \cdots$$ $$F_2(x) = x+ \frac{x^3}{2\cdot3} + \frac{x^5}{2\cdot3\cdot4\cdot5} ...
3
votes
5answers
148 views

Convergent or divergent $\sum_{n=1}^{\infty}\frac{1}{\sqrt{n^2+1}}\left(\frac{n}{n+1}\right)^n$?

Any suggestions? I have tried using D'Alembert's test, but on the end I get 1. I can't think of any other series with which to compare it. In my textbook the give the following solution which I don't ...
3
votes
3answers
112 views

Does $\sum_{n=1}^{\infty}\frac{n-1}{n^2}$ converge or diverge?

Is my logic OK? $a_{n}=\frac{n-1}{n^2}$ $\frac{1}{n} \leq b_{n}=\frac{n-\frac{n}{2}}{n^2}=\frac{n}{2n^2}=\frac{1}{2n} \leq a_{n}=\frac{n-1}{n^2}$ and there for the initial series diverges.
6
votes
0answers
159 views

Connection between integral expression and the factorial of infinity

Does the fact that $$\int_{-\infty}^{\infty}\exp\left(-\frac{1}{2}x^2\right)\mathrm{d}x=\sqrt{2\pi}$$ Have something to do with the fact that the regularized factorial of infinity is also ...