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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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> 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 ...
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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 ...
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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 $x_0=x$, $x_1=f(x)$, $...
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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
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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} = ...
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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
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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 $\sum_{n=1}^{\infty}\frac{1}{n}$ ...
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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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1answer
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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
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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 $a_n$ ...
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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_{...
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0answers
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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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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 ...
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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 ...
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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
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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
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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. $$ ...
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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{...
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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
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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 ...
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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!
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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
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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 ...
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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
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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?
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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
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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
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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
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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
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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
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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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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
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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 $\sigma_s(k)$ ...
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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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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!
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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
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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
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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
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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
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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}^{\...
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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
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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 ...
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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}...