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Questions tagged [divergent-series]

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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1answer
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Prove that $ \sum\limits_{n=2}^\infty \frac{a_n}{\ln{n}} $ diverges if $ \sum\limits_{n=1}^\infty a_n $ diverges and $a_n > 0$ for all $n$ [on hold]

Suppose that $ \sum\limits_{n=1}^\infty a_n $ diverges and $a_n > 0$ for all $n$. Why does $ \sum\limits_{n=2}^\infty \frac{a_n}{\ln{n}} $ diverge?
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2answers
28 views

Infinite divergent series - does the convergence of quotient imply convergence of difference?

I have two series with all terms positive, $$\sum_{i=1}^{n}a_i \equiv A_n >0,\;\; \sum_{i=1}^{n}b_i\equiv B_n>0.$$ Each series diverge as $n\to \infty$. We also have $A_n \leq B_n, \forall n$. ...
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3answers
50 views

How can Taylor series diverge?

I understand how to prove a power series diverges, but that seems to contradict simple logic for me. The idea behind constructing a Taylor series is that it is a polynomial that has the same nth ...
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1answer
22 views

question about convergence and divergence of $\sum (n-2)^3\,e^{-n(x+2)}$

I have the problem that I could not see my fails in the my calculations and why does exists just one right way of showing the convergence or divergence of that formula. the first version is the ...
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0answers
12 views

Suppose $k_n\subset[0,\infty)$ with $k_n\to1$ as $n\to\infty$ and $\alpha_n$ is in $[0,1)$ is $\sum\alpha_n(k_n(M+1)-1)<\infty$ where $M>1$

we am working on a mapp to show that the Mann iteration converges. At the end of the computation we came up the series $\sum\alpha_n(k_n(M+1)-1)$, from previous work it is known that the series $\sum\...
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2answers
23 views

Is there any way to simplify this result for the harmonic series?

I tried simplifying the left-hand side(the harmonic series summation formula) and came up with the right-hand side by finding $\frac{1}{r} - \frac{1}{2r} = \frac{1}{2r}$ and then evaluating by ...
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1answer
30 views

Why can't I use Divergence test instead of Limit comparison test? [closed]

$\sum_{n=1}^{\infty}\frac{n}{n^{2}+1}$ or series an = (n/n^2+1) from $n = 0$ to $\infty$. Why can't I use the divergence test here and multiply top and bottom by $\frac{1}{n^{2}}$ and get the limit is ...
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0answers
35 views

If $a_n > 0$, $\sum a_n$ diverges, and let $b_n$ such that $ \frac{b_n}{a_n} \to L$ then, $\frac{\sum b_n}{\sum a_n} \to L$

Shows that if $a_n > 0$, $\sum a_n$ diverges, and let $b_n$ such that $ \frac{b_n}{a_n} \to L$ then, $\frac{\sum b_n}{\sum a_n} \to L$. My attempts: let $\sum^N_{n=1} a_n = S^a_N$, $\sum^N_{n=1} ...
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2answers
72 views

How to show $1+\frac{1}{2^2} + \frac{2^2}{3^3}+\frac{3^3}{4^4}+… $this series divergent

I've searched in google, but from nowhere I got help. I'm reading undergrad series in real analysis. I'm not able to find it out how to show that this series diverges. Please need a proper solution ...
1
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1answer
27 views

Is there a formal way of defining “giving meaning” to divergent expressions? [duplicate]

Calculations in Quantum Field Theory seem to be full of divergent expressions which are evaluated using various tricks. One such example is the sum of all positive integers: $$ \sum^\infty n = -\frac{...
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0answers
14 views

Point of studying divergent series

The question is basically, why do we study divergent series (changed the phrasing to do away with the pessimistic tone to the header)? Are there any topics in any field, not necessarily within ...
1
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1answer
36 views

Limit Comparison Test with upside-down division

I have a series $\sum_{n=1}^{\infty}a_{n}=\sum_{n=1}^{\infty}\frac{\sqrt[2019]{n+2020}}{n^2-2020}$ and I'm looking for a series that converges in order to use the Limit Comparison Test, such as: $\...
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3answers
37 views

Proving an alternating infinite series to be divergent

I am trying to prove $$\sum_{n=0}^\infty \frac{(-4)^{3n}}{5^{n-1}}$$ is a divergent series. I know that it increases as the series progresses regardless of sign, so it must be divergent, but im not ...
2
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1answer
59 views

Where does this proof of convergence fail?

Given the series, $$\sum_{n=1}^{\infty} (-1)^{n}\frac{n}{n+1}$$ I know we can immediately conclude that it is obviously divergent by the divergence test. But I want to know where exactly am I going ...
2
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3answers
90 views

Convergence of the series $\sum^{\infty}_{n=2} \left(\ln\left(\frac{n}{n-1}\right) - \frac{1}{n}\right) $

Does the following series converge or diverge? \begin{equation} \sum^{\infty}_{n=2} \left(\ln\left(\frac{n}{n-1}\right) - \frac{1}{n}\right) \end{equation} I have noticed that each of partial sum ...
0
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2answers
28 views

Prove convergence or divergence: Integral Test

How can I prove that the series $$\sum_{n=1} \frac{1}{(n+1)(\ln(n)+1)}$$ converges or diverges. I have tried the ratio test, which resulted in $\lim\limits_{x \to ∞} \frac{\frac{1}{((n+1)+1)(ln(n+1)...
2
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3answers
56 views

How to evaluate whether summation of $e^n/(ne^n+1)$ diverges or converges?

How to test this summation for divergence or convergence? $$\sum_{n=0}^\infty \frac{e^n}{ne^n+1}$$ Edit: Here is my work, but I got it wrong. I tried using the comparison test. \begin{align*} a_n &...
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1answer
19 views

Dimensional regularization and expansion of gamma function

In my calculations, I used dimensional regularization, i.e. replace $d\rightarrow d-\epsilon$ and calculated the divergent integral. Then, I would like to expand the answer into seriers by $\epsilon$ ...
1
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2answers
72 views

Convergence of series and summation methods for divergent series

I would like to know what is the sum of this series: $$\sum_{k=1}^\infty \frac{1}{1-(-1)^\frac{n}{k}}$$ with $$ n=1, 2, 3, ...$$ In case the previous series is not convergent, I would like to know ...
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2answers
29 views

Sum of a divergent series depends on the way how one performs the summation

Can some one explain the following sentence regarding summation of divergent series? $\text{Sum of a divergent series depends on the way how one performs the summation}.$ What does mean it? Can ...
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0answers
23 views

comparing two decaying sequences

I have two sequences: $p_0, p_0 \rho_1, p_0 \rho_2, ...$ and $q_0, q_0 \gamma_1, q_0 \gamma_2, ...$. Both sequences have infinite number of terms, and both sequences sum to 1 each. All terms ...
2
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1answer
29 views

Contradiction in radii of convergence? Where is my error?

I'm working through Baby Rudin and I came across what seems to me to be a contradiction, but it could be an error on my part. It has to do with radii of convergence of power series. First, let $\{...
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2answers
68 views

How to convert the following sum to a geometric series?

Find $$ \sum_{n = 1}^{\infty} \frac{6 - 2^{2n - 1}}{3^n} $$ There are many ways to find that the limit is divergent, but the question explicitly states the sum must be interpreted as a geometric ...
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1answer
36 views

Is the series $\sum_k\left| \frac{1}{\ln{(k+1)}}\right|^p$ divergent for $p > 1$

Let $p > 1$. Is the series $$\sum_k\left| \frac{1}{\ln{(k+1)}}\right|^p$$ divergent? This is in the context of finding a sequence that goes to zero but is not an element of $l^p$ space. I saw in ...
4
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2answers
267 views

Convergence/Divergence of infinite series

I'm trying to figure out if the following series converges or diverges. I have spent hours on it and can't figure it out. Tried to use the comparison test, Dirichlet, and Abel, but none of it worked. ...
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1answer
42 views

Prove that the logarithm family of sums diverges?

Define $$a_k(n) =\frac{1}{n\log(n)\log(\log(n)) \cdots \log^{k}(n)}$$ Do all of the sums (for a fixed $k \in \mathbb{N}$): $$A_k = \sum_{n} a_k(n)$$ Diverge?
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4answers
77 views

Show $a_n$ is unbounded if $a_n= a_{n-1} \left(1+ \frac{1}{\sqrt n}\right)$ to determine that $a_n$ diverges.

I wish to determine the limit of $a_n$, where we recursively define: $$a_n= a_{n-1} \left(1+ \frac{1}{\sqrt n}\right)$$ where $a_0=1$ I already noticed it is increasing, because $$a_n - a_{n-1}= \...
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0answers
25 views

What does it mean when a series expansion has increasing negative order in epsilon as epsilon approaches 0?

I'm trying to get an order bound on the following integral as $\epsilon \to 0$: $$g(\epsilon) = \int_a^{a+\epsilon} f(\epsilon,v) dv$$ where $f$ is an ugly, but smooth, function when $a\leq |v| <a+...
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1answer
45 views

Divergence for Series with $|a_n| \ge b_n \ge 0$

first of all i hope you get my point since English is not my native. I know the series $$ y = \sum_{k=1}^\infty \frac{1}{k} $$ is divergent. Now I want to check if the series $$x=\sum_{k=1}^\infty ...
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5answers
153 views

$\sum_{n=1}^\infty \frac{n!e^n}{n^{n+ \frac{3}{2}}}$ - any ideas for a simple proof of divergence?

I am looking for a simple proof of divergence for the series: $\sum_{n=1}^\infty \frac{n!e^n}{n^{n+\frac{3}{2}}}$ That's a part of the more general problem: For what values of X is the series $\sum_{...
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1answer
30 views

Divergent condition of a series.

For $s>0$ when the following series will diverge $$\sum_{k=1}\bigg(\frac{2^k}{1+2^k}\bigg)^{2s}.$$
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0answers
23 views

Similar approach at series and integrals and d'Alembert's test.

We have "duality" between series and improper integrals. For example: We know that convergence doesn't depend on "proper part"*: If $\int_a^{\infty} f$ convergent and $b>a \to \int_a^{\infty} f = ...
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1answer
98 views

Divergent infinite series $n!e^n/n^n$ - simpler proof of divergence?

$$ \sum_{n=1}^\infty \frac{n!e^n}{n^n} $$ Where $e$ is Euler's number. Recently on calculus class we were covering convergence tests and my group got stuck with this infinite series. Our calculus ...
2
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3answers
69 views

Does $\sum q_i = \infty$ imply $\sum \log(1+q_i)=\infty$?

My intuition is that the first-order term in the Taylor expansion should dominate the series, if divergent: $$ \log(1+x) = x - \frac { x ^ { 2 } } { 2 } + \frac { x ^ { 3 } } { 3 } - \frac { x ^ { 4 }...
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1answer
52 views

Improper integral convergence $I = \int_{0^+}^{1^-}\frac{\log(x)}{1-x}dx$

I was solving a few problems regarding convergence and divergence when I ran into this one. I tried searching on the internet but couldn't find an exact match to the problem. The task is to determine ...
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0answers
25 views

Regularization of the following sum

Am interested in how one could regularize the following sum $\sum_{m,n = 1, \infty} \sqrt{m^2 + n^2}$. Would preferably want this in the $\epsilon$ expansion regularization as talked below where a ...
2
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1answer
30 views

Divergent and convergent series of positive terms

I recently read an article which contains the following facts :--- Let $\{a_n\}$ be a sequence of positive number such that $\sum_{n=1}^{\infty} a_n$ diverges, so we must have $a_n\sim \frac{1}{n^p}$ ...
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2answers
283 views

Does $(1+\frac12-\frac13) + (\frac14+\frac15-\frac16)+(\frac17+\frac18-\frac19)+\cdots$ converge?

Does the series $$S=\left(1+\frac{1}{2}-\frac{1}{3} \right) + \left(\frac{1}{4}+\frac{1}{5}-\frac{1}{6} \right)+\left(\frac{1}{7}+\frac{1}{8}-\frac{1}{9}\right)+\cdots$$ converge? Here's my attempt ...
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2answers
44 views

Find radius of convergence of the power series.

Find the radius of convergence of power series $$ \sum_{n=0}^{\infty} 2^{2n} x^{n^2}$$ A)1 B)2 C) 4 D)1/4 I try to apply ratio and root test ( Cauchy–Hadamard theorem ) .but they ...
0
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1answer
47 views

How to tell if series terminates (Legendre ODE)

when solving for the coefficients for the Legendre ODE $(1-x^2)y’’-2xy’+l(l+1)y=0$, I understand how to obtain the recurrence relation $$a_{k+2}=\frac{k(k+1)-l(l+1)}{(k+2)(k+1)}a_k.$$ What I do not ...
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3answers
82 views

If $\sum_{n=1}^{\infty} a_n$ converges, does $\sum_{n=1}^{\infty}\frac{1}{a_n}$ diverge?

If a series $\sum_{n=1}^{\infty} a_n$ converges, does $\sum_{n=1}^{\infty} \dfrac{1}{a_n}$ diverge to infinity?
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0answers
59 views

Equality of perturbed Ramanujan's sum and -1/12

Consider the series $\sum_{k=1}^n k e^{-\varepsilon k}\cos(\varepsilon k)$. If one lets $\varepsilon$ be small enough (for example 0.0001) and $n$ large enough (for example 1.000.000), one sees by ...
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2answers
48 views

Radius of convergence for three series

I need to find the radius of convergence of the 3 following series, but there are no solutions, so I don't know if my steps are correct. 1) $\sum_{n=0}^{\infty}x^{n!}$ 2) $\sum_{n=0}^{\infty} \frac{...
0
votes
1answer
27 views

Indefinite Sum Extension of a Finite Sum Equality

The other night I was considering the way in which we can split a finite sum of any arithmetic function into two finite sums, one for it's odd and another for even index terms : $$\sum _{k=1}^{n} \...
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vote
7answers
128 views

Does $\sum\limits_{n=0}^\infty \frac{e^n\sin n}{n}$ converge or diverge? [closed]

The comparison test does not work on this, so I'm stuck trying to find a way to prove that it diverges. I know it definitely diverges. Any solutions?
3
votes
2answers
612 views

How does this series diverge by limit comparison test?

How does this series diverge by limit comparison test? $$\sum_{n=1}^\infty \sqrt{\frac{n+4}{n^4+4}}$$ I origionally tried using $\frac{1}{n^2}$ for the comparison, but I'm pretty sure it has to be $\...
2
votes
1answer
99 views

Is there a connection between $\zeta(-1)$ and Ramanujan's calculation of the sum over $\mathbb{N}$?

Let me elaborate a little on the matter that I've been mulling over for a little while. This essentially concerns the summation of $1+2+3+...$, how it equals $-1/12$ (in a certain sense, obviously not ...
6
votes
2answers
71 views

Does $\sum_{n=1}^{\infty} \frac{3+(-1)^n}{n}$ converge or diverge?

I'm having trouble figuring out if the following series converges or diverges. $$\sum_{n=1}^{\infty} \frac{3+(-1)^n}{n}$$ Here's my thinking: $$\frac{2}{n} \leq \frac{3+(-1)^n}{n}$$ Since $\sum_{n=...
0
votes
0answers
22 views

Is the (x-a) format necessary when finding the radius of convergence for a geometric series?

So I am taking AP Calculus BC, and we are currently working on convergence and divergence of series. I came across the following problem in one of my homework assignments: Here is the work I did to ...
1
vote
3answers
57 views

Does the series $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{3n+n(-1)^n}$ converge?

I have to find out if the series $$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{3n+n(-1)^n}$$ converges. Root test and ratio test did not work out for me. I also tried the alternating series test, but I can ...