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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.

2
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2answers
495 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 $\...
4
votes
1answer
45 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 ...
3
votes
1answer
62 views

Convergence or divergence of series $\sum_{n=0}^{\infty} \dfrac{n! e^n}{n^n}$

Define convergence or divergence of series $\sum\limits_{n=0}^{\infty} \dfrac{n! e^n}{n^n}$. I used direct, limit comparison test, ratio test (D'alembert's theorem), root test (Cauchy's theorem). But ...
6
votes
2answers
47 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
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0answers
21 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
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3answers
51 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 ...
2
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2answers
66 views

Why doesn't $\sum \frac{sin(\frac{1}{n})}{\sqrt(n)}$ diverge?

Why doesn't $\sum \frac{sin(\frac{1}{n})}{\sqrt(n)}$ diverge? I know that it converges, I just want to know what i'm doing wrong here. Here's what I did. $\sum\frac{-1}{\sqrt n} < $ $\sum \...
0
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0answers
35 views

This semi-harmonic-series converges [duplicate]

We know that the series $\sum_{k=1}^{\infty}\frac{1}{k}$ diverges. The series $\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{8}+\frac{1}{10}+\frac{1}{11}+\dots+\frac{1}{18}+\frac{1}{20}+\...
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2answers
17 views

Converging series and converging alternative series implies absolute convergence?

It is known that that harmonic series diverges, but the alternating form of the harmonic series converges. However, I am not sure if there are examples of series $a_n$ that $$ \sum_{n=1}^{\infty} ...
6
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0answers
82 views

Why do the Borwein integrals stop being $\frac{\pi}{2}$?

I just received the book "single digits - In praise of Small Numbers" by Marc Chamberland. In this book, he gives an interesting integral $$\displaystyle \int_0^\infty \dfrac{\sin x}{x} = \dfrac{\pi}...
0
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1answer
22 views

Radius of Finite Convergence

I am trying to solve a problem where I have to find the radius of finite convergence problem. I believe that I solved the problem correctly, receiving an answer of 1. However, I was informed that this ...
0
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1answer
51 views

Alternating Series Doubts

I've some doubts on Alternating series and Alternating test series. I was trying to clear my mind by practicing with some of my book exercises and found troubles with this one: $$\sum_{n=0}^\infty \...
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0answers
41 views

Applications/examples of these properties?

Here are two interesting properties on series : The first one : Let $(u_n)\in(\mathbb{R^+})^{\mathbb{N}}$ such that $\sum \limits_{n\ge 0} u_n=+\infty$. Then there exists $(v_n)\in(\mathbb{R^+})^{\...
0
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2answers
43 views

Infinite Series Diverges By Divergence Test But Converges By Limit Comparison Test

Image of My Work I understand why this infinite series diverges by the divergence test but I can't find fault in my limit comparison test which says it diverges. Please help. Thanks P.S. if my ...
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1answer
29 views

How to prove when this series of polynomials is convergent and when it is divergent?

Let $P$ and $Q$ be polynomials of degree $k$ and $m$ and suppose $Q(n)\ne0\forall n\in\mathbb N$. Prove that $\sum_{n=1}^\infty\frac{P(n)}{Q(n)}$ is convergent if $m\ge k+2$ and divergent if $m\le k+...
1
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1answer
38 views

Determine whether the series is divergent or convergent and find its sum.

Any tips on how to start this would be great. I'm aware its not geometric but the answer key indicates it diverges, however I have no idea how to physically show that with this equation. $$\sum_{n=0}^\...
1
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3answers
43 views

Prove that a sequence is divergent (By definition - Epsilon-N Way)

First, this is the question: Prove (using epsilon-N definition) that the sequence $ a_n = \left<\sqrt{n}\right> $ is divergent. Note: $ \left<x\right> = x- \lfloor x \rfloor$ My ...
1
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1answer
35 views

If $\big\{n(k);k\in \mathbb{N}\big\}$ be set of all natural numbers none of whose digits is $6$, does $\frac1{n(1)}+\frac1{n(2)}+\ldots$ converge? [duplicate]

If $\big\{n(k);k\in \mathbb{N}\big\}$ be set of all natural numbers none of whose digits is $6$, does $$\frac1{n(1)}+\frac1{n(2)}+\ldots+\frac1{n(k)}+\ldots$$ converge? Actually, I have tried of ...
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1answer
39 views

Does this series diverge or converge?

Does this serie diverge or converge ? Why ? $$\sum \frac{\sqrt{n^3}+e^{-n}}{\sqrt{n^5}+\pi}$$
1
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1answer
39 views

Is analytic continuation well-defined as a summation method?

I am not well versed in summation methods or complex analysis, so I will be presenting a detailed view of my question with examples to illustrate my point as well as a few guiding questions that got ...
0
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1answer
26 views

Convergence of a series involving trigonometric ratios

How do I go about investigating the series $$ \sum _{n=1}^{\infty }\left(1+\frac{2n^{2}+n}{3n^3-n\sin (n)}\right)^{\tfrac{n^2+n\cos(n)}{n+\sin(n)}} $$ for convergence or divergence? What I don't ...
1
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1answer
63 views

(Proof Verification) Examining whether the series $\sum \limits_{n=0}^\infty \dfrac{(-1)^{n+1}}{5n+1}$ is convergent, absolute convergent or divergent

Everything in red is edited To show, that the series is convergent we show at first, that $\color{red}{\lim \limits_{n \to \infty} \left(\dfrac{1}{5n+1}\right)}=0$. $\color{red}{\lim \limits_{n \to \...
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votes
1answer
21 views

How to determine whether the following two infinite series converge absolutely, converge conditionally, or diverge. [closed]

I need some guidance on how to solve these, I'm not understanding series and sequences too well and I need an explanation that hasn't come from my lecturer. $$\sum_{k=1}^\infty \frac{\log k}{k^2}$$ $...
2
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4answers
78 views

I was told that I couldn't “pull the limit in”. Tell me exactly how I'm messing up, please!

So, the problem that we were solving was $$\lim_{n\to\infty}\left(\frac{n}{n+1}\right)^n$$ To figure out whether the series converged or diverged, after simplification, I asked my professor whether ...
0
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0answers
47 views

Is there an option to handle Neumann-series when it diverges? (using infinite-sized Carleman matrices)

Motivation I'm considering functions, represented by Carleman matrices of infinite size, for instance $f(x)=t^x - 1$. Let us denote the iterates of $f(x)$ by indexes on $x$ like $x_0=x$, $x_1=f(x)$, $...
5
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0answers
62 views

What is a common framework for these divergent sums?

If you expand $2^x$ using a finite difference series you end up with the formula $$ 1 + x + \frac{1}{2!}x(x-1) + \frac{1}{3!}x(x-1)(x-2) ... = \sum_{n=0}^{\infty} \frac{(x)_n}{n!} $$ Now these ...
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0answers
42 views

A question about divergent series related to the harmonic series

I was thinking about slowly diverging series. I came up with this idea but I don't know what it's called or where to look. Suppose you start with the harmonic series: $$\sum_1^{\infty}\frac{1}{n} = ...
0
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1answer
24 views

Given the series (-1)^n.tan(1/n) how do I study its nature in terms of divergence and convergence?

I have a series whose general term is tan(1/n)*(-1)^n and I want know if it is divergent or convergent, how do I proceed? I have tried stablishing upper and lower limits and the ratio test and all I ...
5
votes
1answer
133 views

On convergence of sums of the form $\sum_{n=1}^{\infty}\frac{1}{n^{1+f(n)}}$

The p-series convergence test is a classic and well-known result for sums of the form $\sum_{n=1}^{\infty}\frac{1}{n^p}$ for a real number $p$. It is known that $\sum_{n=1}^{\infty}\frac{1}{n}$ ...
0
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1answer
60 views

show that $\frac {a_{N+1}}{s_{N+1}} +…+\frac {a_{N+k}}{s_{N+k}}\geq 1 -\frac {a_{N}}{s_{N+k}} $ [duplicate]

Suggestion of how to do it, please. Suppose $\{a_n\}$ is a succession in $\mathbb R ^+$ such that $\sum a_n$ diverges, and if $s_n = \sum\limits_{k=1}^n{a_k}$. show that $\frac {a_{N+1}}{s_{N+...
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1answer
34 views

why this statement about $\sum_{i=0}^n a_n$ is false?

Given that ${a_n}$ is positive series, and $\sum_{i=0}^n a_n$ is converge: -There is a sub-series for ${a_n}$ that converge to S>$0$. (THIS statement is false) why is this statement false?
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1answer
18 views

Determining convergence of series using “Direct comparison test”

Given that $\sum a_n\leq \sum b_n$, if the series of $b_n$ converges, so does the series of $a_n$. In my opinion this idea seems to be very general, because it is easy to find $b_n$ bigger than $a_n$ ...
8
votes
6answers
611 views

Convergence of $\sum_{n=1}^{\infty}e^{-\sqrt{n}}$ using the integral test

Given the series : $$\sum_{n=1}^{\infty}e^{-\sqrt{n}}$$ Determine if convergent or divergent. The function is positive and monotonically decreasing function so I've used the "Integral Test" $$\int_{...
1
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0answers
32 views

Show that $\sum_{n=1}^{\infty}\sin \left(\frac{n\pi}{3} \right)\frac{1}{n^r}$ diverges for $0<r<1$

I would like to show that $\displaystyle \sum_{n=1}^{\infty}\sin \left(\frac{n\pi}{3} \right)\frac{1}{n^r}$ diverges when $0<r<1$. I'm having a hard time doing this though. It seems that p-...
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vote
0answers
54 views

Why can't the divergence of the harmonic series be resolved?

Calculus 1 students are taught that some infinite series converge while others do not. However, there are other tricks one can do whereby some of those "diverging" series do in fact converge but in ...
1
vote
1answer
41 views

For which $p$'s does $\sum _{n\in \mathbb{N}}\Bigl(\frac{1}{\sqrt{n}\log(1+n)}\Bigr)^{p}$ converge

For which $p's \in \mathbb{R}_\geq1$ does the series $\sum _{n\in \mathbb{N}}\Bigl(\frac{1}{\sqrt{n}\log(1+n)}\Bigr)^{p}$ converge Thoughts Clearly the sequence $\frac{1}{\sqrt{n}\log(1+n)}$ tends to ...
0
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0answers
34 views

Harmonic series (Maths and Music)

I am a high school student trying to apply calculus to music (harmonic series). I am just wondering, how can I collect data from any online music app (with music tones that form harmonic series - ...
2
votes
2answers
61 views

Does the Shannon Entropy always exist (even for infinite distributions)?

Let $p : \mathbb{N} \to [0, 1]$ be a probability distribution over the naturals. The Shannon Entropy is: $$H = -\sum_{n=0}^\infty p(n)\log_2 p(n)$$ Does this series always converge? I tried a ...
2
votes
2answers
47 views

Divergence of infinite series $(\frac{3k-2}{4k+2})^{2k-3}$

I would like to ask if my solution for testing the divergence of the infinite series below is correct. $$\sum_{k=1}^{\infty} \left(\frac{3k-2}{4k+2}\right)^{2k-3}$$ I used the Cauchy ratio test. $$ ...
0
votes
1answer
48 views

Show that for a.e. $x\in[0,1]$, $\sum_{n=1}^\infty\sum_{k=1}^n\dfrac{1}{n^\gamma}\dfrac{1}{\sqrt{|x-\frac{k}{2^n}|}}$ converges.

One point to add, $\gamma>2$. My approach is to show its complement, i.e $x$ where the series is divergent, is measure zero by Borel-Cantelli lemma. Let $A_{j,m}:=\{\sum_{n=1}^m\sum_{k=1}^n\dfrac{...
1
vote
1answer
59 views

Do the partial sums of a divergent series converge to Cesaro or Abel sums in some metric?

Let $(a_n)$ be a sequence in $\mathbb{R}$, and let $s_n$ be the $n^{th}$ partial sum of the sequence. Then the Cesaro sum of $(a_n)$ is the limit of the average of the first $n$ partial sums as $n$ ...
3
votes
1answer
53 views

How would I go about determining whether this series converges or diverges?

$$\sum_{n=1}^{\infty} (\ln(2(n+1))- \ln(2n))$$ I was able to plug this into a calculator to determine that the series is divergent. I also graphed the series to observe a decreasing, continuous ...
0
votes
0answers
11 views

Finding the partial sum of a non-geometric/arithmetic series for arbitrarily large numbers

What would be a good way to approximate the partial sum of something similar to $\sum_{n=1e50}^{5e50} n^{2.5}$ ? First time posting in a while!
6
votes
5answers
261 views

How to determine if this series converges or diverges [closed]

How can you determine if this series converges or diverges? $$\sum_{n=1}^{\infty} \frac{(n)^{n^2}}{(n+1)^{n^2}}$$
3
votes
1answer
64 views

Closed expressions for divergent series over Bernoulli numbers?

Motivation In a recent post (Asymptotic behaviour of sums involving $k$, $\log(k)$ and $H_{k}$) I asked for the asymptotic behaviour of the sum $$\sigma_{c}(n)=\sum_{k=1}^n H_{k} \log(k)\tag{1}$$ ...
3
votes
0answers
31 views

regularizing function under sqrt [closed]

Is it possible to find finite part of $\Sigma^\infty_n\sqrt{n^2 +a^2}$ using something like regularized zeta function?
0
votes
1answer
50 views

Does the sum $\sum_{k=1}^{\infty}B_{(4k-2)}+B_{(4k)}$ converges?

Just as the title says I'd like to know if this sum $$ \sum_{k=1}^{\infty}(B_{(4k-2)}+B_{(4k)}) $$ converges and if so to which value. Here $B_{2k}$ are Bernoulli numbers. I've tried with Mathematica ...
2
votes
0answers
52 views

Sequences ${a_k}$ and ${c_k}$ such that $a_k$, $c_k \rightarrow 0$, $\sum_k a_k = \infty$, and $\sum_k\left(\frac{a_k}{c_k}\right )^{2} < \infty $

Is there an example of two sequences for which the following conditions hold? Sequences ${a_k}$ and ${c_k}$ such that $a_k$, $c_k \rightarrow 0$, $\sum_k a_k = \infty$, and $\sum_k\left(\frac{a_k}{...
1
vote
1answer
26 views

$\sum_k\left(\frac{a_k}{c_k}\right )^{2}$, $a_k=\frac{1}{k}$, $c_k$=$1$, $\frac{1}{2}$,$\frac{1}{2}$,$\frac{1}{3}$,$\frac{1}{3}$,$\frac{1}{3}$, …

With $a_k$ and $c_k$ (1 repeating once, then 1/2 twice, then 1/3 three times, then 1/4 4 times, etc) as defined above, does the sum converge?
8
votes
4answers
224 views

How to prove the series $\sum\limits_{n=1}^\infty\frac1n\sin(\ln n)$ diverges

How to prove the following series is divergent: $$ \sum_{n=1}^{\infty} \frac{1}{n} \sin(\ln n)\ ? $$ What I was thinking is, since $\sum\limits_{n=1}^{\infty} \frac{1}{n}$ diverges, $\sin$ is ...