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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3answers
44 views

Does the following series converge uniformly?

I know how to show that the following series will converge absolutely. But am unsure how to show it will or will not converge uniformly for $z\in (0,1).$ $\displaystyle \sum_{n \mathop = 1}^{\infty} ...
0
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1answer
17 views

Does the series $\displaystyle\sum_{j=1}^\infty -\log (1-p_j^{-3/4})$ diverge, where $\{p_j\}$ is the set of primes in increasing order?

Here, $\log$ is the natual logarithm. Is there a simple convergence test I can use? Thanks.
3
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4answers
401 views

1+4+10+20+35+…=? [closed]

Is there a finite value to the infinite sum of all the tetrahedral numbers: $$\sum_{n=1}^\infty \frac{n(n+1)(n+2)}{6}.$$ I know it's a divergent series, but I hear that $$ ...
1
vote
2answers
157 views

Proof of the reciprocal of all semiprimes diverging?

$$\sum_{\text{semi-primes}}\frac{1}{s}=\frac{1}{4}+\frac{1}{6}+\frac{1}{9}+\frac{1}{10}\cdots$$ I almost positive that this sum diverges, but I would really like to see a very thorough proof. Thank ...
1
vote
3answers
92 views

If the sum of $f(n)$ diverge, then does the sum of $\sqrt{f(n)}$ diverge?

Lets say that $$\sum_{n=a}^\infty f(n)$$ diverges. Does $$\sum_{n=a}^\infty \sqrt{f(n)}$$ necessarily diverge?
2
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0answers
25 views

Understand and Implement adjoint operator of gradient

I have an equation as With initial ui=vi; yi=0, tau=1; I am implementing eq. 4.19 and 4.20 in MATLAB. Hence, I would like to ask you something. The adjoint operator of gradient is similar the ...
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2answers
72 views

Does $\sum (-1)^k 2^{1/k}$ converge or diverge?

How am I supposed to determine the convergence of this series if I only know about the alternating series test and the divergence test?
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0answers
23 views

List of divergent series and their summations

On the web one can manage to find a lot of lists of convergent series and their summation btw I didn't find (at least on a quick search) a corrispective list of divergent series, does anyone know one ...
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2answers
97 views

convergence of $\sum^\infty_{k=1} \frac {\ln(k)}{k^p}$

For what values of p does $\sum^\infty_{k=1} \frac {\ln(k)}{k^p}$ converge? Here is my work: $\ln (k) < k$ on $[1,\infty)$ so $\frac {\ln (k)}{k^p} < \frac {k}{k^p}$ Therefore, $ ...
0
votes
1answer
29 views

Ramanujan sum of a function which diverges bot at $0$ and $1$

Wikipedia gives the formulas for the Ramanujan summation of a divergent series in the two cases of a function which has no divergence at $x=0$ and at $x=1$ but what to do with a function which is ...
0
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2answers
49 views

Convergence of a product series with one $1/k$ factor

Let $\left( a_n \right)_n$ be a sequence such that $a_n < 1$, $a_n \rightarrow 0$. Prove or disprove (with a counter-example) that $$ \sum_{n=1}^{\infty} \frac{a_n}{n} < \infty.$$ Comments. If ...
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0answers
27 views

Regularization of a (divergent) cosine series

What would be a suitable regularized value for the following divergent series: $$ S(y) = \sum_{k=1}^{\infty} \cos(k y) \quad y \in R\\ $$ By way of added context, this series arises from a formal ...
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7answers
200 views

Sum: $1-2+3-4+5-6+…$

If we forget all the rules about infinte sums what am I doing wrong? $$1-2+3-4+5-6+...=\sum_{n=1}^{\infty} n(-1)^{n+1}$$ (with Grandi's series) $$1,1+(-2)=-1,1+(-2)+3=2,1+(-2)+3+(-4)=-2,...$$ we ...
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4answers
41 views

Should I use the comparison test for the following series?

Given the following series $$\sum_{k=0}^\infty \frac{\sin 2k}{1+2^k}$$ I'm supposed to determine whether it converges or diverges. Am I supposed to use the comparison test for this? My guess would ...
0
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2answers
95 views

Two divergent series such that their product is convergent

I faced a series question it goes something like give an example of 2 divergent series such that when the 2 series are multiplied to each other, the new series becomes convergent, although it looks ...
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3answers
104 views

Sum of $\sum\limits_{x=-\infty}^{\infty}x^{\operatorname{sign}(x)}$

Both the sum of $1+2+3+4+\cdots$ and the sum of $\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\cdots$ diverge. If both are paired together in one function, as seen above, can they amount to a ...
1
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3answers
42 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
45 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 ...
0
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4answers
150 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 ...
2
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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
79 views

Summing divergent asymptotic series [closed]

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 ...
4
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1answer
85 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). $ $ ...
0
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1answer
49 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 ...
0
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1answer
29 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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votes
2answers
43 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 ...
6
votes
4answers
267 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. ...
0
votes
1answer
35 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) ...
1
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2answers
58 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 ...
6
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6answers
111 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: ...
0
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1answer
54 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 ...
0
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1answer
12 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
59 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} ...
13
votes
3answers
581 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: ...
2
votes
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!}} { ...
1
vote
0answers
121 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 ...
1
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1answer
54 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 $ ...
2
votes
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.
0
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2answers
26 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? $$ \sum \limits^{\propto }_{n=1}\frac{(-1)}{n(\ln(n+1))^{2}} $$
0
votes
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.
1
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2answers
90 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 ...
1
vote
1answer
58 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 ...
4
votes
3answers
82 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 ...
0
votes
3answers
102 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
votes
2answers
57 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 $s=0$: ...
1
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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 ...
1
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1answer
29 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 $ $ \ \ \ \ ...
4
votes
2answers
406 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 ...
1
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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 ...
1
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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 ...
3
votes
0answers
44 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 ...