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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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1answer
138 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 ...
1
vote
3answers
61 views

Divergence/convergence of the series

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
42 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}}$...
-2
votes
0answers
80 views

Ramanujan's 1/pi Series: Proving that $a_{n+1} < La_n$ (for $n \geq 1$) implies that $a_n< L^{n-1}a_1$ for $n \geq 2$.

I don't know how to go about this question regarding Ramanujan's formula: $$\frac{1}{\pi} = \sum_{n=0}^{\infty} \frac{\sqrt{8}(4n)!(1103+26390n)}{9801(n!)^4396^{4n}}$$ Let $a_n$ denote the nth term ...
-1
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} \...
5
votes
3answers
59 views

Does $\sum_{n=1}^{\infty}\frac{1}{2^n} + \frac{3}{n}$ converge or diverge?

Does this series converge or diverge? If it converges, determine its limit. $$\sum_{n=1}^{\infty}\frac{1}{2^n} + \frac{3}{n}$$ So far I said that $\frac{1}{2^n}$ is a geomotric series that converges,...
4
votes
0answers
61 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
59 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}^{...
-2
votes
1answer
53 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
52 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
166 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 ...
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votes
4answers
76 views

Is the series $\sum_{k=1}^\infty \frac{1}{k^{3/2}} $ convergent or divergent? [closed]

$\sum_{k=1}^\infty \frac{1}{k^{3/2}} $ should be between the harmonic series $\sum_{k=1}^\infty \frac{1}{k} $ which diverges and $\sum_{k=1}^\infty \frac{1}{k^2} $ which converges. So I am not ...
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
82 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 ...
1
vote
1answer
68 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. click here for the image ...
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0answers
49 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
119 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
114 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
42 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
144 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
156 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}...
1
vote
1answer
55 views

which of the following series convergent?

which of the following series converges? A. $\sum_\limits{n=1}^\infty \frac{1}{3^{\ln n}}$ B. $\sum_\limits{n=1}^\infty \frac{1}{2^{\ln n}}$ C. $\sum_\limits{n=1}^\infty \frac{1}{n^{1+ \frac{1}{n} }...
0
votes
1answer
40 views

Showing $\sum_{n=2}^{\infty} \frac{1}{n \ln(n^3)}$ diverges [duplicate]

Two questions: Show whether $\sum_{n=2}^{\infty} \frac{1}{n \ln(n^3)}$ converges or diverges. According to wolfram, the series diverges by the comparison test, so I tried the following: for $n$ ...
1
vote
1answer
38 views

True or false ? Abel theorem in the complex case

Let $a_k $ be complex numbers such that the radius of convergence of $\sum a_k z^k$ is $1$. Suppose $|\sum_{k=0}^{N} a_k |\longrightarrow +\infty $ as $N\to +\infty$. Then, is it true that $$\...
0
votes
0answers
114 views

Series solution of Legendre differential equation

For the Legendre differential equation $$(1-x^2)y''-2xy'+l(l+1)y=0$$ The solution is $$y=a_0\left[1-\frac{l(l+1)}{2!}x^2+\frac{l(l+1)(l-2)(l+3)}{4!}x^4\right] + a_1\left[x-\frac{(l-1)(l+2)}{3!}x^3+...
-1
votes
1answer
213 views

Is this considered a disproof of $1+2+3+4\ldots=-\frac{1}{12}$? [closed]

The series "$1+2+3+4\ldots=-\frac{1}{12}$" didn't seem to make sense to me as it breaks clear rules of series (but I am yet to research if the rules it broke aren't breakable(doesn't work if broken) ...
0
votes
1answer
49 views

Abel's theorem infinite case

I would like to prove the second remark in the Wikipedia article of Abel's theorem: Let $a_k $ be real numbers. If $\sum_{k=0}^{\infty} a_k = +\infty $ then $$\lim_{z\to 1^-} \sum_{k=0}^{\infty} a_k ...
0
votes
1answer
28 views

expanding logarithm in series

In Boas it's stated that $$ln(p+y\sqrt{p})$$ can be expanded as $$ln(p+y\sqrt{p}) = ln(p)+ln\left(1+\frac{y}{\sqrt{p}}\right) = ln(p)+\frac{y}{\sqrt{p}}-\frac{y^2}{2p}$$ At first I tried to expand ...
1
vote
1answer
71 views

Find the sum of the series: $\sum_{n=5}^{\infty} \frac{6}{n^2 - 3n}$

In terms of finding the sum of the series $\sum_{n=5}^{\infty} \frac{6}{n^2 - 3n}$ , this series looks to me like it is telescoping. So, I tried to factor it in a way to find a telescoping pattern, ...
0
votes
1answer
60 views

Characteristic function for a random variable that can take the value infinity

I want to derive a characteristic function for the duration of a stochastic process that can possibly never end. Specifically, I have $X \in \mathbb{N} \cup \{0, +\infty \}$ and $\sum_{k = 0}^{\infty} ...
1
vote
1answer
45 views

Is the sum of the reciprocals of Ramanujan primes divergent?

I have read many a wonderful proof that the sum of the reciprocals of the primes is divergent and I know that the sum of the reciprocals of twin primes does not diverge, but do we know any results ...
0
votes
1answer
29 views

Give an example which proves/disproves the condition for convergence of the series.

Firstly I have these 2 conditions for arbitrary sequences {$x_n$}: Given any 1$\lt$q$\lt$p$\lt$ $\infty$, there is a real sequence {$x_n$} such that $\sum_{n=1}^{\infty}|{x_n}|^{p}$ is convergent ...
3
votes
1answer
41 views

Growing slower than exponential imply some divergence

Is the following true? Let $(x_n)_{\mathbb{N}}$ be a sequence such that for all $c > 1$, exists $m_c \in \mathbb{N}$ whith $x_n < c^n, \forall n > m_c$. Then exists $\gamma > 0$, for ...
0
votes
4answers
45 views

Question on Series Convergence

Suppose we have the series $$\sum\limits_{n=1}^{\infty} \frac{n+2}{(n+1)\sqrt{n+3}}$$ and want to test for convergence. I have tried a number of things--ratio test, various comparison tests, ...
0
votes
1answer
18 views

Relationship between divergence of these two series

Take a sequence of non-negative numbers $(t_k)_{k=1}^\infty$ with $\lim\limits_{k\to\infty}t_k=0$ (the sequence is not necessarily monotonously decreasing). I'm interested in the relationship between ...
3
votes
1answer
123 views

Divergent series which is Abel summable but not Euler summable

It is said that: Abel summation and Euler summation are not comparable. We were able to find examples of divergent series which are Euler summable but not Abel summable, for instance $$ 1-2+4-8+...
0
votes
1answer
31 views

Limit of a sequence with real and natural number variables

Currently learning about limits and I'd like to check out if I calculated that all correctly: number 1: SOLVED number 2: UNSOLVED $$a_n=\frac{c_2n^2+c_1n+c_0}{b_2n^2+b_1n+b_0} \text{with $c_1,c_2,...
2
votes
2answers
143 views

Prove that the series $\sum_{n=1}^\infty \frac{e^nn!}{n^n}$ diverges

Prove that the series diverges: $$\sum_{n=1}^\infty \frac{e^nn!}{n^n}$$ I've tried using the ratio test but after all of my calculations I got that $\lim_{x\to\infty}\frac{a_{n+1}}{a_n} = 1$, which I ...
-1
votes
2answers
38 views

Summation series question divergent inequality

where n is > 0 and an integer k(n) = ln(n) $$h(n) =\sum_{i=1}^n \frac{1}{i}$$ prove h(n) > k(n) ive tried and considered that h(n) is a divergent series so when n approaches infinity, im not sure ...
0
votes
1answer
53 views

Cesàro summation and operations

We are reading now about Cesàro summation. And there is a remark that: We want $(a_n)$ and $(b_n)$ Cesàro-summable, then the Cauchy product $$(c_n)=\sum_{k=0}^na_kb_{n-k}$$ is Cesàro-summable....
1
vote
1answer
86 views

Finding roots of trigonometric function

I have been given the following functions (where $\omega_0$ is some positive constant) $$\begin{align} \Gamma(t) &= \frac{1}{t^2} \left(\; \frac{1}{2} \omega_0^2 t^2 - \cos(\omega_0 t) - \...
0
votes
0answers
29 views

Sum of reciprocals of prime numbers, using alternating sings

Does the Sum from N=1 to N=∞ of : (-1^(N+1))((1/P(N)), where P(N) is the nth prime number, converge? If so, to what number does it converge? Or in different terms, does the sum of the reciprocal of ...
2
votes
5answers
62 views

Calculate the following convergent series: $\sum _{n=1}^{\infty }\:\frac{1}{n\left(n+3\right)}$

I need to tell if a following series convergent and if so, find it's value: $$ \sum _{n=1}^{\infty }\:\frac{1}{n\left(n+3\right)} $$ I've noticed that $$ \sum _{n=1}^{\infty }\:\frac{1}{n\left(n+3\...
1
vote
2answers
52 views

Inequality about the most common divergent series [closed]

We know that: $1+\frac{1}{2}+\frac{1}{3}+....$ is a divergent series. I have a small problem about this series. Show that: $1+\frac{1}{2}+\frac{1}{3}+....+\frac{1}{2^{2006}} >50$ How to prove ...
0
votes
1answer
33 views

For which values of $\gamma>0$ do we have $\lim_{n \to \infty} \frac{1}{n^2}\sum_{k=1}^n k^\gamma=0$

I am interested in getting to know for which values of $\gamma >0$ we have that $$ \lim_{n \to \infty} \frac{1}{n^2}\sum_{k=1}^n k^\gamma <\infty $$ Or more specifically, for which values $\...
0
votes
1answer
17 views

Notation for a divergent series

When something converge we use the notation $<\infty$. For example For all $s$ $s.t$ $Re(s)>1$ $$\zeta(s)<\infty$$ Can we say the opposite for a divergent series? i.e. $$\sum_{n=1}^\...
1
vote
2answers
29 views

Valid Series Convergence Proof?

Just wanted someone to check me to see if this is a valid proof. I've seen other proofs, but I can't use Cauchy condensation, or integral test as we haven't learned it yet. I have ratio test and ...
-1
votes
1answer
34 views

Determine whether the series is convergent or divergent

$$\sum_{n=0}^{\infty} \frac{1}{x^n}$$ $$ \sum_{n=0}^{\infty} \frac{1}{x^n}= \sum_{n=0}^{\infty} \: \left( \frac{1}{x} \right)^n = \left(\frac{1}{x}\right)^n=\frac{a}{1-r} = \frac{x}{x-1} \: when \: |x|...