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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Divergence series as a sum of convergent and divergent series

Suppose $S_n = \sum_{k=1}^n a_k,$ where $a_k \geq 0.$ We know that $S_n \to \infty,$ when $n \to \infty.$ I wonder if it is possible to write $S_n = \sum_{k=1}^{n/2} a_k + \sum^{n}_{i=n/2} a_k,$ such ...
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4answers
32 views

Looking at slow divergent series.

So today i have two questions in one, basically i need explanations. It is school break and where can i find a better place to tutor myself with math apart from here. Now I came across this topic of ...
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0answers
34 views

Interpretations of divergent series

I have already bumped into this question and found it not quite useful for my own purposes. The usual way the sum of the infinite series is taught and defined in 1st year undergrad course is that its ...
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4answers
143 views

How to derive $\sum_{n=0}^\infty 1 = -\frac{1}{2}$ without zeta regularization

On Wikipedia we find $\displaystyle \bbox[5px,border:1px solid #F5A029]{1 + 1 + 1+\dots =\sum_{n=0}^\infty 1 = -\frac{1}{2}}$ using (the rather complicated) zeta-function regularization. I asking for ...
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0answers
9 views

Is there a readable diagram illustrating all the Tauberian-type theorems?

The Wikipedia page Divergent Series lists dozens of various methods for "summing" divergent series, without any real indication of the relations between them. Is there anywhere I could find a ...
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2answers
78 views

Summing divergent asymptotic series [on hold]

I found the sine integral si to be $$Si (x)\sim \frac \pi 2+\sum _{n=1}^\infty (-1)^n \left(\frac{(2 n-1)! \sin (x)}{x^{2 n}}+\frac{(2 n-2)! \cos (x)}{x^{2 n-1}}\right)$$ Say I want to find ...
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1answer
80 views

Simplify P(n), where n is a positive integer : $ P(x)=\sum \limits_{k=1}^\infty \arctan\left(\frac{x-1}{(k+x+1)\sqrt{k+1}+(k+2)\sqrt{k+x}}\right). $

This is what I have tried, but I don't know what to do next, so I need help : $ P(x)=\sum \limits_{k=1}^\infty \arctan\left(\frac{x-1}{(n+x+1)\sqrt{n+1}+(n+2)\sqrt{n+x}}\right). $ $ ...
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1answer
42 views

$\sum 0$: does it converge or diverge?

Sometimes I have to do exercise with parameter and, if I substitue particular value of the parameter, I obtain $\sum_{n=1}^{\infty} 0$. But it isn't clear for me if in this case the series converges ...
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1answer
24 views

Convergence of Series for tangent (only convergence or divergence)

$$\sum_{n=17}^{\infty}\left(\tan\left(\frac{1}{n}\right)\right)^2 \ \ $$ My first guess is to write the series as integral. And use the substitution for u=1/n. That changes my upper and lower ...
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2answers
40 views

Complex infinity when proving divergence

My calculus course book (Adams' Calculus) does not explain why $(-1)^n$ diverges (it just says "$(-1)^n$ simply diverges"), and I tried to see why it diverges by taking its limit as $n$ approaches ...
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261 views

When will it diverge? When will it converge?

Test for what $x\in \mathbb{R}$ the series $\sum_{n=0}^\infty nx^n$ converges and for what $x\in \mathbb{R}$ it diverges. Determine the limit of sequence for the case of the convergence. ...
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1answer
33 views

Ratio test, Root test, and Divergence test related.

(I) Ratio test: If the result is smaller than 1 then the sum is convergent, and if the sum is larger than 1 then the sum is divergent, and that got me thinking if negative infinity (smaller than 1) ...
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2answers
57 views

How to solve this sequence?

I have this sequence: $\sum_{n=1}^{\infty} \frac{n^2+n-1}{\sqrt{n^\alpha+n+3}}$ For which values of $\alpha$ does this converge? I first tried to separate into cases where $\alpha \gt 0$ etc and ...
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6answers
106 views

Does the series: $\sum_{n=1}^\infty \frac{2n+1}{n^2(n+1)^2}$ converge?

does $\sum_{n=1}^\infty \frac{2n+1}{n^2(n+1)^2}$ converge? I think yes, it does, because the $a_n$ in the series converges to zero. but I'm trying to prove this by the help of the fact that: ...
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1answer
52 views

Does the series: $\sum_{n=1}^\infty (-1)^n \lbrack {\sqrt\frac{n}{2}} \rbrack$ Converge?

Does the series: $\sum_{n=1}^\infty (-1)^n \lbrack {\sqrt\frac{n}{2}} \rbrack$ Converge ? Note: by the brackets I mean the floor function. I tried to substitute numbers and look at the members of ...
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1answer
10 views

The majorant/minorant criterion

The majorant criterion says if a series in a Banach space has a convergent majorant, then it converges absolutely. My question is, what if a series in a Banach space has a convergent minorant, does it ...
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3answers
58 views

Convergence of series $\sum \limits^{\infty }_{n=1}\frac{n^{(n+\frac{1}{n} )}}{(n+\frac{1}{n} )^{\frac{1}{n} }}$

i need help for find method or methods for solve this series and find the convergence. I very appreciate for any help and yours comments. $$\sum \limits^{\infty }_{n=1}\frac{n^{(n+\frac{1}{n} ...
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3answers
565 views

Limit approach to finding $1+2+3+4+\ldots$

When exploring the divergent series consisting of the sum of all natural numbers $$\sum_{k=1}^\infty k=1+2+3+4+\ldots$$ I came across the following identity involving a one-sided limit: ...
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4answers
47 views

Using the Limit Comparison Test on $\sum_{n=1}^{\infty} \frac{n^2} {n!}$

is this right ? $$ \sum_{n=1}^{\infty} \frac{n^2} {n!} $$ i need to use quotient criterion $$ \lim_{n\to\infty} \frac{\frac{n^2}{n!}}{ \frac{1}{n!}} = \lim_{n\to\infty} {\frac{n^2}{n!}} { ...
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0answers
108 views

Zeta regulated product, solving without the zeta function.

Earlier i've asked about how to calculate divergent products, i got some directions which made me curious. Now i'm wondering is this correctly done. Divergent products. The most commen divergent ...
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1answer
52 views

Absolute convergence of $\sum a_n$

I would ask a help for the following problem If someone could tell me what criteria or applies so I would appreciate. Show that if $ \sum \limits^{\infty }_{n=1}a_n $ is absolutely convergent, then $ ...
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2answers
63 views

Convergence of $\frac{1}{(\ln n)^{\ln n}}$

Could I have a hint for testing the convergence of the following series please? $$\sum_{n=2}^\infty\frac{1}{(\ln n)^{\ln n}}$$ I am very appreciative for your help.
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2answers
25 views

Prove absolute convergence from alternants

He failed to get the show in its entirety in this series, one I could indicate how working with this kind of series? $$ \sum \limits^{\propto }_{n=1}\frac{(-1)}{n(\ln(n+1))^{2}} $$
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1answer
36 views

How prove that $\sum_{j=1}^{\infty} \prod_{k=1}^j \frac{k-1.5}{k} = -1$ [closed]

How prove that $$\sum_{j=1}^{\infty} \prod_{k=1}^j \frac{k-1.5}{k} = -1$$ I have any idea, so any help wil be helpfull.
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2answers
87 views

Why $\zeta(-2) $ is not $\sum_{n=1}^{\infty}\frac{1}{n^{-2}}$? [duplicate]

Let $\zeta(s)= \sum_{n=1}^{\infty}\frac{1}{n^{s}}$ a standard formula. I'm confused if you tell me: does this series: $\sum_{n=1}^{\infty}\frac{1} {n^{s}}$ converge? I will answer you: this series ...
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1answer
47 views

Harmonic Series and Its Divergence by Abel Sum and Cesaro Sum

Already I know that harmonic series, $$\sum_{k=1}^n\frac1k $$ is divergent series. And, it is also divergent by Abel Sum or Cesaro Sum. However, I do not know how to prove it is divergent by concept ...
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3answers
81 views

Proof of $\sum_{x = 1}^\infty \frac{1}{x}$'s divergence by absurdity?

(From this site.) The following argument purports to show that the series $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} \dots = 0$. It begins with the harmonic series. $$ \begin{aligned} \sum ...
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99 views

Find whether $\int_{1}^{\infty} \frac{sin(x)}{x} dx$ is converging or not

Find whether $\int_{1}^{\infty} \frac{sin(x)}{x} dx$ is converging or not. I tried to use comparison test or limit comparison test but could't find a suitable function. How can I determine what type ...
3
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2answers
56 views

Regularizing the $\log\log n$ series

The divergent series $$\sum_{n=1}^\infty\log n$$ can be regularized using the derivative of the Riemann zeta function at $s=0$: ...
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0answers
36 views

Limit comparison test how to choose $b_n$?

$$\sum_{n=1}^\infty \frac{2n-1}{4n^2+1}\tag{1}$$ i would like to find out if this series convergent or not so i use Limit comparison test and choose $a_n$ and $b_n=\frac{1}{n}$ why do i need to ...
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1answer
25 views

What are the basic rules for manipulating diverging infinite series?

This is something that I played around with in Calc II, and it really confuses me: $s = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + \ldots = \infty$ $s - s = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + \ldots $ $ \ \ \ \ ...
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2answers
399 views

Does this sum converge, is my solution good?

$$ \sum_{n=1}^\infty \frac{\sin(n)^{7}}{(n^{7}+1)^{1/2}} $$ I would say that it doesn't converge, cause I would write this as: $$ $$ $$ \frac{\sin(n)^{7}}{(n^{7})^{1/2}} $$ when $$ \lim_{n\to ...
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2answers
40 views

Convergence of series 4

Determine if the following series is convergent or not: $$\frac{1}{\sqrt{n} \log n}$$ I tried: $a_k = \frac{1}{\sqrt{n} \log n}$ $b_k= \frac{1}{\sqrt{n}}$ then did: $\frac{a_k}{b_k}$ and got ...
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2answers
57 views

Show that this series is divergent using the comparison test.

Let $\langle a_n\rangle$ be a sequence of positive numbers. Consider $\sum_{n=1}^\infty{Y_n}=\sum_{n=1}^\infty{\left(\frac{a_n}{n}+\frac{n}{a_n^2}+\frac{a_n}{n^3}\right)}$. Show this diverges using ...
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0answers
43 views

Elemenatry topological proof of Erdos conjecture on Arithmetic Sequences

Define $C_n(A) = \{a \in A : \forall d \in \Bbb{N}$ one of $a + d, a +2d, \dots a + (n-1)d$ is not in $A\}$ For $n \leq m$, we have $C_n(A) \subset C_m(A)$. Proof. Let $x$ be in the LHS. Then ...
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0answers
34 views

Fibonacci Sequence (conceptual/mechanical)

I'm in Calc II and have my exam on series/sequences tomorrow. There will be an extra credit question related to Fibonacci and am trying to gather as much info on the sequence as possible. -If the ...
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0answers
18 views

Expression about $\sum_{d=0}^{\infty} e^{dt}\frac{(5d)!}{(d!)^5}5\sum_{n=0}^{\infty} \left( \frac{1}{n+d+1} - \frac{1}{n+5d+1} \right)$ [Done]

The purpose of my question in (here)[converge value of series $\sum_{n=0}^{\infty} \left( \frac{1}{n+d+1} - \frac{1}{n+5d+1} \right) $ was actually to re-expression of following computation. ...
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1answer
58 views

converge value of series $\sum_{n=0}^{\infty} \left( \frac{1}{n+d+1} - \frac{1}{n+5d+1} \right) $

\begin{align} \sum_{n=0}^{\infty} \left( \frac{1}{n+d+1} - \frac{1}{n+5d+1} \right) = \sum_{n=0}^{\infty}\frac{4d}{(n+d+1)(n+5d+1)}= ? \end{align} I know from the $p$-test, ($i.e$ $\sum \frac{1}{n^p}$ ...
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2answers
117 views

In what sense does $\sum_{k=0}^{\infty} 2^{2k} = - {1 \over 3}$?

In The Road to Reality Penrose remarks on an identity written down by Euler which is "obviously wrong" and yet correct "on some deeper level". He makes reference to the series again when discussing ...
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1answer
46 views

Does this summation (involving binomial) have a closed form? If so, what is it?

The following sums are the ones I'm interested in: $\sum_{i=m}^{\Omega}{\binom{i}{m}i^{-k}}$ $\lim_{\Omega\rightarrow\infty}{\sum_{i=m}^{\Omega}{\binom{i}{m}i^{-k}}}$ I already know that ...
2
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1answer
51 views

Is the sum of the reciprocals of the squarefree numbers divergent or convergent?

I just come across this question by trying to analyze the pseudoinverse of some infinite matrix (the matrix T as interpreted in my answer to this MSE-question), where this series occurs from some ...
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1answer
41 views

Is the following series convergent or divergent. If convergent find the sum if possible. [closed]

How do you find if this series is convergent or divergent? $$\sum_{n=1}^{\infty} \dfrac{\ln(n^2+1)}{\ln(n^3+1)}$$ Is someone able to solve this with all the steps?
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1answer
60 views

Partial sum of all natural numbers [duplicate]

I'm pretty sure this question has been asked several times however,... It's obvious that the series $S_n = 1+2+3+4+\cdots +n$ is divergent. However there are many other claims that the sum is -1/12 ...
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1answer
38 views

$\sum a_{n}$ is convergent, $\sum a_{n}^2$ is divergent. Prove $\sum a_{n}$ is conditionally convergent.

Suppose that the series $\sum_{n=1}^{\infty}a_{n}$ converges while $\sum_{n=1}^{\infty}a_{n}^2$ diverges. Prove that $\sum_{n=1}^{\infty}a_{n}$ converges conditionally.
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1answer
48 views

Proof of divergence of a series 4

Let $ (a_k)_{k\in\mathbb{N}}$ be a decreasing sequence of positive real numbers. We suppose that there exists a $b>0$ such that $a_k \geq \frac{b}{k}$ for infinite values of $k$ . Prove that the ...
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2answers
72 views

Proof infinite series $1 + 2 + 3 +\cdots +n+\cdots$ diverges

Question: By considering the partial sums for S, that is $S_n = 1 + 2 + 3 +\cdots +n$ show that the infinite series S does not converge. My answer : I tried to attempt this question, ...
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5answers
139 views

How do you solve for the limit of this series?

I need to take the limit of this summation so that I kind find out whether it converges or diverges. The equation is: $$\sum_{k=1}^\infty \frac{4}{k+4}$$ What I have tried so far is the following: ...
0
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2answers
52 views

Convergence of $\sum_{n=1}^{\infty} \frac{n^3}{\ln(2)^n}$

would the sum from 1 to infinity of $\frac{n^3}{(\ln{2})^n}$ converge? In the limit n tends to infinity the denominator grows more quickly and so the terms go to zero. Using the ratio test I get ...
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1answer
30 views

Discuss whether the series $\sum \left[(\pi/2)^a - (\arctan n)^a\right]$ converges or not, based on the value of $a$

$$\sum_{n=1}^\infty {\left[ {{{\left( {\frac{\pi }{2}} \right)}^a} - {{(\arctan n)}^a}} \right]} $$ I proved that the series diverges for $a < 0 $ and that the series converges for $a = 1$ (using ...
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2answers
64 views

Divergent, convergent series

Let $p$, $q \in \mathbb{R}$ and see the series $$ \sum_{n=2}^{\infty} \frac{1}{n^p(\ln n)^q} $$ View with the comparison criterion that if $p> 1$ then the series is convergent for all $q$, and ...