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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0answers
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> If {$a_{n}$} is divergent, can {$|a_{n}|$} be convergent? > If {$a_{n}$} is convergent, can {$a^2_{n}$} be divergent? [on hold]
Can anyone help me prove these two problems?
If {$a_{n}$} is divergent, can {$|a_{n}|$} be convergent?
If {$a_{n}$} is convergent, can {$a^2_{n}$} be divergent?
Thank you
I´ve seen an ...
2
votes
4answers
69 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
43 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
57 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
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0answers
40 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:
$$\sum_1^{\infty}\frac{1}{n} = ...
0
votes
1answer
23 views
Given the series (-1)^n.tan(1/n) how do I study its nature in terms of divergence and convergence?
I have a series whose general term is tan(1/n)*(-1)^n and I want know if it is divergent or convergent, how do I proceed?
I have tried stablishing upper and lower limits and the ratio test and all I ...
5
votes
1answer
130 views
On convergence of sums of the form $\sum_{n=1}^{\infty}\frac{1}{n^{1+f(n)}}$
The p-series convergence test is a classic and well-known result for sums of the form $\sum_{n=1}^{\infty}\frac{1}{n^p}$ for a real number $p$. It is known that $\sum_{n=1}^{\infty}\frac{1}{n}$ ...
0
votes
1answer
53 views
show that $\frac {a_{N+1}}{s_{N+1}} +…+\frac {a_{N+k}}{s_{N+k}}\geq 1 -\frac {a_{N}}{s_{N+k}} $ [duplicate]
Suggestion of how to do it, please.
Suppose $\{a_n\}$ is a succession in $\mathbb R ^+$ such that $\sum a_n$ diverges, and if $s_n = \sum\limits_{k=1}^n{a_k}$.
show that $\frac {a_{N+1}}{s_{N+...
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votes
1answer
31 views
why this statement about $\sum_{i=0}^n a_n$ is false?
Given that ${a_n}$ is positive series, and $\sum_{i=0}^n a_n$ is converge:
-There is a sub-series for ${a_n}$ that converge to S>$0$. (THIS statement is false)
why is this statement false?
0
votes
1answer
18 views
Determining convergence of series using “Direct comparison test”
Given that $\sum a_n\leq \sum b_n$, if the series of $b_n$ converges, so does the series of $a_n$.
In my opinion this idea seems to be very general, because it is easy to find $b_n$ bigger than $a_n$ ...
8
votes
6answers
601 views
Convergence of $\sum_{n=1}^{\infty}e^{-\sqrt{n}}$ using the integral test
Given the series : $$\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
31 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
47 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
40 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
26 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
52 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
44 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.
$$
...
0
votes
1answer
41 views
Show that for a.e. $x\in[0,1]$, $\sum_{n=1}^\infty\sum_{k=1}^n\dfrac{1}{n^\gamma}\dfrac{1}{\sqrt{|x-\frac{k}{2^n}|}}$ converges.
One point to add, $\gamma>2$.
My approach is to show its complement, i.e $x$ where the series is divergent, is measure zero by Borel-Cantelli lemma.
Let $A_{j,m}:=\{\sum_{n=1}^m\sum_{k=1}^n\dfrac{...
1
vote
1answer
54 views
Do the partial sums of a divergent series converge to Cesaro or Abel sums in some metric?
Let $(a_n)$ be a sequence in $\mathbb{R}$, and let $s_n$ be the $n^{th}$ partial sum of the sequence. Then the Cesaro sum of $(a_n)$ is the limit of the average of the first $n$ partial sums as $n$ ...
3
votes
1answer
53 views
How would I go about determining whether this series converges or diverges?
$$\sum_{n=1}^{\infty} (\ln(2(n+1))- \ln(2n))$$
I was able to plug this into a calculator to determine that the series is divergent. I also graphed the series to observe a decreasing, continuous ...
0
votes
0answers
10 views
Finding the partial sum of a non-geometric/arithmetic series for arbitrarily large numbers
What would be a good way to approximate the partial sum of something similar to $\sum_{n=1e50}^{5e50} n^{2.5}$ ?
First time posting in a while!
6
votes
5answers
242 views
How to determine if this series converges or diverges [closed]
How can you determine if this series converges or diverges?
$$\sum_{n=1}^{\infty} \frac{(n)^{n^2}}{(n+1)^{n^2}}$$
3
votes
1answer
58 views
Closed expressions for divergent series over Bernoulli numbers?
Motivation
In a recent post (Asymptotic behaviour of sums involving $k$, $\log(k)$ and $H_{k}$) I asked for the asymptotic behaviour of the sum
$$\sigma_{c}(n)=\sum_{k=1}^n H_{k} \log(k)\tag{1}$$
...
2
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0answers
28 views
regularizing function under sqrt [closed]
Is it possible to find finite part of $\Sigma^\infty_n\sqrt{n^2 +a^2}$ using something like regularized zeta function?
0
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1answer
50 views
Does the sum $\sum_{k=1}^{\infty}B_{(4k-2)}+B_{(4k)}$ converges?
Just as the title says I'd like to know if this sum
$$
\sum_{k=1}^{\infty}(B_{(4k-2)}+B_{(4k)})
$$
converges and if so to which value. Here $B_{2k}$ are Bernoulli numbers.
I've tried with Mathematica ...
2
votes
0answers
51 views
Sequences ${a_k}$ and ${c_k}$ such that $a_k$, $c_k \rightarrow 0$, $\sum_k a_k = \infty$, and $\sum_k\left(\frac{a_k}{c_k}\right )^{2} < \infty $
Is there an example of two sequences for which the following conditions hold?
Sequences ${a_k}$ and ${c_k}$ such that $a_k$, $c_k \rightarrow 0$, $\sum_k a_k = \infty$, and $\sum_k\left(\frac{a_k}{...
1
vote
1answer
26 views
$\sum_k\left(\frac{a_k}{c_k}\right )^{2}$, $a_k=\frac{1}{k}$, $c_k$=$1$, $\frac{1}{2}$,$\frac{1}{2}$,$\frac{1}{3}$,$\frac{1}{3}$,$\frac{1}{3}$, …
With $a_k$ and $c_k$ (1 repeating once, then 1/2 twice, then 1/3 three times, then 1/4 4 times, etc) as defined above, does the sum converge?
7
votes
4answers
186 views
How to prove the series $\sum\limits_{n=1}^\infty\frac1n\sin(\ln n)$ diverges
How to prove the following series is divergent:
$$
\sum_{n=1}^{\infty} \frac{1}{n} \sin(\ln n)\ ?
$$
What I was thinking is, since $\sum\limits_{n=1}^{\infty} \frac{1}{n}$ diverges, $\sin$ is ...
10
votes
1answer
129 views
Function $f$ s.t. $\lim_{x\to\infty}\frac{f(e^x)}{f(x)}=1$
The questions are:
1) Does there exists some function $f$ s.t.
$\lim_{x\to\infty}\frac{f(e^x)}{f(x)}=1$ and
$\lim_{x\to\infty}f(x)=\infty$?
2) Is $\big(\sum_{k=n}^{2^n}a_k\big)_n\to0$ is ...
4
votes
5answers
89 views
Nature of infinite series $ \sum\limits_{n\geq 1}\left[\frac{1}{n} - \log(1 + \frac{1}{n})\right] $
$$\sum\limits_{n\geq 1}\left[\frac{1}{n} - \log\left(1 + \frac{1}{n}\right)\right]$$
Is it convergent or divergent?
Wolfram suggests to use comparison test but I can't find an auxiliary series.
2
votes
1answer
154 views
Convergence/Divergence of an Infinite Series with Natural Logarithms
I've spent a good week and half manipulating and trying different tests to find the convergence or divergence of this series:
$$\sum_{n=0}^\infty \frac{1}{(\ln n)^{\ln n}}$$
I've tried all the ...
0
votes
3answers
69 views
Divergence/convergence of the series [closed]
Given that $\frac{1}{\sqrt{x}} \ge \frac{1}{x+1}$ for all $x> 0$, determine the convergence or divergence of the series,
$$\sum_{k=1}^{\infty}\left( \frac{1}{\sqrt{2k-1}} - \frac{1}{2k} \right)$$
...
0
votes
2answers
44 views
What should I use to find the convergence of the series $n\arctan\frac{1}{n^3}$?
I thought of using comparison test and used the other series as $V_n = \frac{1}{n^2}$ . Now using the limit comparsion rule and LH rule to evaluate limit $$\frac{\arctan(\frac{1}{n^3})}{\frac{1}{n^3}}$...
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votes
1answer
56 views
Evaluating Sum of $\dfrac{i}{(-x)^i}$ [duplicate]
I would like to ask if the expression below can be simplified using standard summation properties? Or should I dive into much deeper concepts like the power series? Thank you.
$$\sum_{i=1}^{n-1} \...
4
votes
0answers
67 views
Approximations to series of Ramanujan-type
Recently I have been playing around with series of the form
$$\sum_{k=1}^{\infty}\frac{k^{s}}{e^{kz}-1} = \sum_{k=1}^{\infty}\sigma_{s}(k)e^{-kz}$$
for $s \in \mathbb{Z}$ and where $\sigma_s(k)$ ...
0
votes
3answers
60 views
Convergent Series (Rudin)
Rudin proves a statement in his classical book. Most of this proof is straightforward, while a subtle point confuses me.
Claim: Suppose $a_1 \geq a_2 \geq \dots \geq 0$. Then, the series $\sum_{n=1}^{...
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votes
1answer
59 views
Question concerning sigma notation [duplicate]
Consider you have been given that
$$\sum_{i = 1}^{\infty}i = -\dfrac{1}{12} $$
How do you solve this sigma notation? I've not seen this kinda sigma notation before.
Regards!
0
votes
0answers
53 views
Laurent series of an integral with parameter
To find the Laurent series of function $f(a)$ at point $a=0$
$$
f(a)=\int^1_0 \frac{d x}{x^2+a^2}
$$
one can first do the integral
$$
f(a)=\frac{1}{a}\arctan(1/a)
$$
then expand $\arctan(1/a)$ and ...
5
votes
1answer
95 views
Sequence formed by quotient of sum and the product of odd squares is divergent: Why?
Trust me when I tell you this is not homework. Can you suggest an solid argument to prove
$$ S_n = \frac{1}{3^25^27^2\cdots (2n-1)^2}\sum_{m=3}^{\infty} 2^{(2n)m}e^{-2^{m/2}}
$$
diverges as $n \...
3
votes
2answers
195 views
How do I derive the Ramanujan Summation of $\sum_{n=1}^{\infty}n^2 = 0$?
I'm sure everyone has seen the infamous identity of $\sum_{n=1}^{\infty}n^k=\frac{-1}{12}$, when $k=1$, and likely the associated series manipulations used to get that. I'm attempting to do a similar ...
1
vote
1answer
51 views
Finding the asymptotic behavior of a function series
I solved the shape of an elastic sheet annulus clamped on the inner and outer circle with a point load (the figure below shows the cross section of an example): Solve Green function of an annulus to ...
2
votes
4answers
87 views
Does $\sum_{n=1}^{\infty}\arctan(\frac{1}{n^2})$ converge?
I tried to prove that the series $\sum_{n=1}^{\infty}\arctan(\frac{1}{n^2})$ converges. I tried using $\lim_{n\to \infty}\frac{\arctan(\frac{1}{n^2})}{\frac{1}{n^2}}$ as we knew that $\sum_{n=1}^{\...
1
vote
1answer
32 views
Radius Convergence
$$\sum \frac{(-1)^nz^{2n+1}}{\log n} $$
I'm stuck on this problem. I tried the ratio test and the root test and i keep getting that the the series diverges. Am i doing something wrong? Any info will ...
2
votes
1answer
90 views
how to get $\sum_{k=1}^\infty \arctan\biggr(\frac{10k}{(3k^2+2)(9k^2-1)}\biggr)=\log3-\frac{\pi}{4}$
problem in the above asked equation of S.Ramanujan !
Hello everyone,this is a result of an entry described by ramanujan,i first request you to see the photo i have attached.
MY ATTEMPTION
From LHS ...
1
vote
0answers
50 views
Alternating series that is bounded by 1 but fails an alternating series test
I have the following recursion of sequence $\{A_i\}_{i=1}^\infty$ and $\{B_i\}_{i=0}^\infty$
Index
1 $~~~~~~~~~~~~~~A_1=q\sigma$ $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~B_1=B_0-A_1B_0/\sigma$
2 $~~~~~~~~~~...
4
votes
0answers
124 views
New/useful method for summation of divergent series?
Questions
$$ S(n,x) = x+e^x + e^{e^x} + e^{e^{e^x}} + \dots \text{$n$ times}$$
Also obeys (see background for argument):
$$ \frac{1}{2 \pi i} \oint e^{S(k,x)} \frac{\partial \ln(\frac{\int_0^\...
7
votes
0answers
129 views
Calculus of variation with discontinuous solutions
I'm thinking of the following question:
Consider a function $f: [0,L]\rightarrow\mathbb{R}$ and an energy functional $$F=\int_{0}^{L}\Big (\frac{\mathrm{d}f}{\mathrm{d}x}\Big)^2\mathrm{d}x.$$ The ...
1
vote
2answers
47 views
Leibnitz series? $\sum\limits_{n=1}^{\infty} (-1)^{n} \frac{n^{2} +3n - \sin(n)}{n^{4}-\arctan(n^{2})}$
Good evening everyone,
I'd like to discuss with you the following exercise :
$\sum\limits_{n=1}^{\infty} (-1)^{n} \frac{n^{2} +3n - \sin(n)}{n^{4}-\arctan(n^{2})}$
I can prove that $\lim\limits_{x \...
1
vote
2answers
146 views
How to prove this question by Ramanujan?
click here for photo
$$1+2\sum_{k=1}^\infty \frac{1}{(4k)^3-(4k)}= \frac{3}{2}\ln(2)\,.$$
well i have attatched a photo which has been asked to prove without using calculus,but how to solve this ...
4
votes
5answers
166 views
Bounds for the Harmonic k-th partial sum.
I need to bound the k-th partial sum or the Harmonic series. i.e.
$$ln(k+1)<\sum_{m=1}^{k}\frac{1}{m}<1+ln(k)$$
I'm triying to integrate in $[m,m+1]$ in the relation $\frac{1}{m+1}<\frac{1}...
