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
votes
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
votes
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
vote
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
votes
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
votes
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}+\...
0
votes
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
$$ \sum_{n=1}^{\infty} ...
6
votes
0answers
82 views
Why do the Borwein integrals stop being $\frac{\pi}{2}$?
I just received the book "single digits - In praise of Small Numbers" by Marc Chamberland.
In this book, he gives an interesting integral
$$\displaystyle \int_0^\infty \dfrac{\sin x}{x} = \dfrac{\pi}...
0
votes
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
votes
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:
$$\sum_{n=0}^\infty \...
0
votes
0answers
41 views
Applications/examples of these properties?
Here are two interesting properties on series :
The first one :
Let $(u_n)\in(\mathbb{R^+})^{\mathbb{N}}$ such that $\sum \limits_{n\ge 0} u_n=+\infty$. Then there exists $(v_n)\in(\mathbb{R^+})^{\...
0
votes
2answers
43 views
Infinite Series Diverges By Divergence Test But Converges By Limit Comparison Test
Image of My Work
I understand why this infinite series diverges by the divergence test but I can't find fault in my limit comparison test which says it diverges. Please help. Thanks
P.S. if my ...
0
votes
1answer
29 views
How to prove when this series of polynomials is convergent and when it is divergent?
Let $P$ and $Q$ be polynomials of degree $k$ and $m$ and suppose $Q(n)\ne0\forall n\in\mathbb N$.
Prove that $\sum_{n=1}^\infty\frac{P(n)}{Q(n)}$ is convergent if $m\ge k+2$ and divergent if $m\le k+...
1
vote
1answer
38 views
Determine whether the series is divergent or convergent and find its sum.
Any tips on how to start this would be great. I'm aware its not geometric but the answer key indicates it diverges, however I have no idea how to physically show that with this equation.
$$\sum_{n=0}^\...
1
vote
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
vote
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 ...
0
votes
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
vote
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
votes
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
vote
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 \...
-1
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
votes
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
votes
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
votes
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 ...
0
votes
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:
$$\sum_1^{\infty}\frac{1}{n} = ...
0
votes
1answer
24 views
Given the series (-1)^n.tan(1/n) how do I study its nature in terms of divergence and convergence?
I have a series whose general term is tan(1/n)*(-1)^n and I want know if it is divergent or convergent, how do I proceed?
I have tried stablishing upper and lower limits and the ratio test and all I ...
5
votes
1answer
133 views
On convergence of sums of the form $\sum_{n=1}^{\infty}\frac{1}{n^{1+f(n)}}$
The p-series convergence test is a classic and well-known result for sums of the form $\sum_{n=1}^{\infty}\frac{1}{n^p}$ for a real number $p$. It is known that $\sum_{n=1}^{\infty}\frac{1}{n}$ ...
0
votes
1answer
60 views
show that $\frac {a_{N+1}}{s_{N+1}} +…+\frac {a_{N+k}}{s_{N+k}}\geq 1 -\frac {a_{N}}{s_{N+k}} $ [duplicate]
Suggestion of how to do it, please.
Suppose $\{a_n\}$ is a succession in $\mathbb R ^+$ such that $\sum a_n$ diverges, and if $s_n = \sum\limits_{k=1}^n{a_k}$.
show that $\frac {a_{N+1}}{s_{N+...
-1
votes
1answer
34 views
why this statement about $\sum_{i=0}^n a_n$ is false?
Given that ${a_n}$ is positive series, and $\sum_{i=0}^n a_n$ is converge:
-There is a sub-series for ${a_n}$ that converge to S>$0$. (THIS statement is false)
why is this statement false?
0
votes
1answer
18 views
Determining convergence of series using “Direct comparison test”
Given that $\sum a_n\leq \sum b_n$, if the series of $b_n$ converges, so does the series of $a_n$.
In my opinion this idea seems to be very general, because it is easy to find $b_n$ bigger than $a_n$ ...
8
votes
6answers
611 views
Convergence of $\sum_{n=1}^{\infty}e^{-\sqrt{n}}$ using the integral test
Given the series : $$\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
vote
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-...
1
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
votes
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 ...
