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.

Filter by
Sorted by
Tagged with
0 votes
1 answer
47 views

Determine $f_k(n)$ in $\sum_{i=0}^n \left (1-\frac{1}{2^i} \right )^k - f_k(n) \rightarrow 0$

For a given $k \in \mathbb{N}$, could you determine $f_k(n)$ such that the following holds? $$\sum_{i=0}^n \left (1-\frac{1}{2^i} \right )^k - f_k(n) \rightarrow 0$$ For $k=1$, $$ \sum_{i=0}^n \left (...
Amir's user avatar
  • 493
-2 votes
0 answers
25 views

Determine $\sum_{n=1}^\infty {1/S_n}$ where $S_n=\sum_ {k=1}^n {1/\ln(k+1)}$

Let $S_n=\sum_{k=1}^n {1/\ln(k+1)};$ then $\sum_{n=1}^{\infty} {1/S_n}$ diverges? In fact, I used the Stolz-Cesaro theorem to find out that it diverges, but I do not completely understand the theorem ...
김석훈's user avatar
1 vote
1 answer
82 views

How can I show that the given series diverges?

How can I show that this series diverges? $$ \sum_{n=1}^{\infty} \frac{\bigl(2 + (-1)^n\bigr)^n}{\sqrt{n}\,3^n} $$ The standard tests kinda fail in this case. I am not getting any clue on how to ...
Confusedphysica's user avatar
0 votes
2 answers
68 views

Asymptotic evaluation of a series

I am reading a paper in which the authors derive the asymptotics for the following series: $$ \sum_{k=0;\,k+=2}^{N(1+m)-2}\left(1-\frac{k}{2}\log{\frac{k+2}{k}}\right)\sim \frac{1}{2}\log{\left(\pi N(...
Giancarlo Creanza's user avatar
0 votes
0 answers
24 views

If $\sum\frac{1}{a_n}$ diverges, then $\sum\frac{1}{n\max_{1\leq k\leq n}\frac{a_{n+1} - a_k}{n-k+1}}$ diverges.

Let $(a_n)$ be a strictly increasing sequence of positive real numbers, and denote $\Delta a_n:= a_{n+1} - a_n.$ We know that if $\displaystyle\sum_{n\in \mathbb{N}} \frac{1}{a_n}$ diverges, then $\...
Adam Rubinson's user avatar
2 votes
0 answers
54 views

Is my proof of the divergence of prime reciprocals valid

I tried to prove the divergence of the prime reciprocals as a challenge and I think I came up with quite an intuitive argument using Borell Cantelli, but maybe not rigorous. For two primes $p_n>...
AndroidBeginner's user avatar
2 votes
1 answer
76 views

Proofs involving manipulation of divergent series

Is this proof valid even though the harmonic series it is based on is divergent? Prove: $$\sum_{n=2}^\infty (\zeta(n)-1)= 1$$ Where $\zeta$ as in Riemann's Zeta function is summed over all natural ...
Older Amateur's user avatar
0 votes
1 answer
46 views

Numerically compute an oscillating series

I would like to compute in a numerically stable way an oscillating series. Imagine I have a signal $C(n)$, $n\in\mathbb{N}$ which decays exponentially with $n$ e.g. $C(n) = e^{-2n}$. Also, imagine I ...
Matteo Saccardi's user avatar
1 vote
0 answers
27 views

Convergence of derived series

I found this question on another forum so I don't have more details about it other than it seems non-intuitive to me: Let $\sum_{n=1}^\infty a_n$ consist of positive terms and be convergent. Define $...
Ivan's user avatar
  • 2,304
1 vote
1 answer
74 views

Does this imaginary series really diverge? [duplicate]

$$\sum_{n=2}^{\infty}{\frac{(-i)^{n}}{\ln n}}$$ In the answer, using comparison test $1/\ln n > 1/n,$ the series is divergent. But, in my opinion, the series can be divided like $$\sum_{n=1}^{\...
J sw's user avatar
  • 21
-2 votes
1 answer
100 views

How to evaluate $\sum \frac{1}{2^{n-r}}\frac{1}{n}$? [closed]

$$\sum_{n=r+1}^{\infty}\frac{1}{2^{n-r}}\frac{1}{n}$$ ,where r is a positive integer. First, I wanna know if this is convergent(I guess so) and WHY. And if so, how to evaluate it?
David Lee's user avatar
  • 131
0 votes
0 answers
10 views

Concoct an alternating series which diverges that is alternating, the limit goes to 0, but is NOT always decreasing. [duplicate]

Basically need an example of a divergent series that is alternating and the limit goes to 0.
Super walrus's user avatar
1 vote
0 answers
37 views

Divergence of Mehler's Hermite polynomial series

According to Mehler's formula, $$ \sum\limits_{n=0}^{\infty}\left(\cfrac{\rho}{2}\right)^n\cfrac{H_n(x)H_n(y)}{n!}=\cfrac{1}{\sqrt{1-\rho^2}}\exp\left(-\cfrac{\rho^2(x^2+y^2)-2\rho x y}{1-\rho^2}\...
george_ch's user avatar
1 vote
0 answers
55 views

Can a function be represented as 2 different power series

Find the power series representation for 𝑓(𝑥)=1/x+2 cant i write the power series as $$ f(x) = \frac{1}{x+2}. $$ so I am a little confused because cant I write the series as $$ f(x) = \frac{1}{1-(-x-...
barış yaycı's user avatar
0 votes
1 answer
36 views

Question about convergence and divergence of positive series,which means there exists no slowest convergent series.

If positive series $\sum_{n=1}^{\infty} a_n$ is divergent,try to prove exist divergence positive series $\sum_{n=1}^{\infty} b_n$,which satisfies $\lim _{n \rightarrow \infty} \frac{b_n}{a_n}=0$ If ...
Dropsy Zheng's user avatar
1 vote
1 answer
91 views

T\F: if $\lim_{k\to\infty}\sum_{i=n_k}^{n_{k+1}-1}|\alpha_i|= 0$ then $\sum \alpha_k$ converges.

Let $\sum \alpha_k$ be a bounded series of reals. Suppose $\{n_k\}_{k=1}^{\infty}$ is a strictly-increasing monotonic sequence of naturals s.t $lim_{k\to\infty} (n_{k+1}-n_{k})= \infty$. Prove or ...
X4J's user avatar
  • 963
3 votes
1 answer
117 views

If $n\in\mathbb N$, then $\sum_{k=1}^{\infty} \frac{1}{k^n}$ diverges. What's wrong in my proof?

If $n\in\mathbb N$, then $\sum_{k=1}^{\infty} \frac{1}{k^n}$ diverges. Clearly this is false in general and only holds for $n=1$ . But here's my wrong proof in which I can't find what's wrong. Let $$...
An_Elephant's user avatar
  • 2,459
1 vote
1 answer
75 views

Determining radius of convergence of the series $\displaystyle{\sum_{n=0}^{\infty} \sin(nz^n)}$

I have the following question: for what values of $z\in \mathbb{C}$ does the following series converges: $\displaystyle{\sum_{n=0}^{\infty} \sin(nz^n)}$ My initial idea was to use the series expansion ...
obitobi_tobias's user avatar
1 vote
1 answer
38 views

How to judge the positive series$\frac{\left[2+(-1)^n\right]^n}{n}\left(\frac{1}{3}\right)^n$ when comparision/root test don't work?

Comparision/root test don't work for the serise$\frac{\left[2+(-1)^n\right]^n}{n}\left(\frac{1}{3}\right)^n$
高小辉's user avatar
1 vote
2 answers
70 views

Is there a conclusive convergence test for $\sum_{n=1}^\infty (e^{2/n^2}-1)$?

I have tried testing for convergence using the common methods but none of them were conclusive. Wolfram Alpha tells me $$\sum_{n=1}^\infty \left (e^{2/n^2}-1\right )$$ does approximately converge to ...
IO.SYS's user avatar
  • 35
9 votes
1 answer
269 views

Suppose $\sum \frac{1}{a_n} $ diverges. Then does $\sum \frac{1}{n\Delta a_n}\ $ diverge?

Let $(a_n)$ be a strictly increasing sequence of positive real numbers, and denote $\Delta a_n:= a_{n+1} - a_n.$ Suppose $\displaystyle\sum \frac{1}{a_n} $ diverges. Then does $\displaystyle\sum \frac{...
Adam Rubinson's user avatar
6 votes
0 answers
78 views

Is the series $ \sum_{n=1}^{\infty} \frac{\cos (\frac{1}{n^2})}{\sqrt{2n+1}}$ divergent or convergent?

I want to know if my reasoning is correct about the divergence of the series $\displaystyle \sum_{n=1}^{\infty} \frac{\cos (\frac{1}{n^2})}{\sqrt{2n+1}}$. First, we know that $\displaystyle \frac{1}{n^...
LegendCero's user avatar
1 vote
0 answers
53 views

Is this proof that $\sum_{n=1}^{\infty} \sin(n)\frac{(2^n+5^n)(n+1)^2}{n!}+\frac{\cos(1/n)}{\sqrt n}$ diverges correct?

Does the following series converge absolutely, converge conditionally or diverge: $$\sum_{n=1}^{\infty} \sin(n)\frac{(2^n+5^n)(n+1)^2}{n!}+\frac{\cos(1/n)}{\sqrt n}.$$ Let $$a_n=\sin(n)\frac{(2^n+5^n)...
two's user avatar
  • 900
2 votes
2 answers
72 views

Is $\sum_{n=2}^{\infty}\sin {\left(n\pi +\frac{1}{n^{\alpha}\log n}\right)}, \alpha<0$ divergent?

I konw $\sum_{n=2}^{\infty}\sin {\left(n\pi +\frac{1}{n^{\alpha}\log n}\right)}=\sum_{n=2}^{\infty}(-1)^n\sin {\left( \frac{1}{n^{\alpha}\log n}\right)},$ if I could prove $\sin{\left( \frac{1}{n^{\...
Ychen's user avatar
  • 551
4 votes
1 answer
193 views

Help me ask this confused question about the divergence of the harmonic series

I've struggled for a couple of weeks to figure out the question I really want to ask, but without any success, so I'm just putting out this rather confused question out in hopes that someone can help ...
MJD's user avatar
  • 64.9k
2 votes
2 answers
130 views

Serie of max of two values diverges implies that the max of the series diverges.

Context : In the context of simplifying the conditions presented in this post, the following question arise. Question : Suppose that $0<a_k,b_k$ for $k\in\mathbb N$ are such that $\sum_{k\geq 1} \...
P. Quinton's user avatar
  • 5,517
2 votes
1 answer
192 views

Has this weak version of Erdős Conjecture on arithmetic progressions been proven, or is it still an open problem?

This question is motivated by Erdős conjecture on arithmetic progressions. It is a weaker version of Erdős Conjecture, but I do not know how to prove it. Erdős conjecture on arithmetic progressions ...
Adam Rubinson's user avatar
3 votes
0 answers
107 views

Does $\sum\limits_{k=1}^n\dfrac{1}{k^{1+|\sin k|}}$ approach $a\ln (\ln n)$?

$\sum\limits_{k=1}^{\infty}\dfrac{1}{k^{1+|\sin k|}}$ has been shown to diverge. This made me wonder about the asymptotics of $S(n)=\sum\limits_{k=1}^n\dfrac{1}{k^{1+|\sin k|}}$. Numerical ...
Dan's user avatar
  • 16.3k
0 votes
2 answers
87 views

Is 1/n gradually approaching 0 when n approaches infinity.

According to the definition of Convergent series, "the series is said to be Convergent when n approaches infinite an=L, where L is a constant." And when we prove that n/(n+1) will approach 1,...
Ili a's user avatar
  • 19
1 vote
0 answers
29 views

Series convergence value

Find a real number $A>0$ such that $\sum_{n=1}^\infty \frac{n!}{(cn)^n}$ converges if $c > A$ and diverges if $c < A$. I started by performing the ratio test. From that I deduced that $c$ ...
cindy's user avatar
  • 129
0 votes
1 answer
72 views

Asymptotic expansion for reciprocal of Bessel function

The asymptotic expansion for the Bessel function $J_n(x)$ with integer $n$ is known and given at https://dlmf.nist.gov/10.17. Is there a way to find the asymptotic expansion for $1/J_n(x)$? I'm not ...
gb62442's user avatar
  • 25
1 vote
1 answer
54 views

Imaginary part in asymptotics of hypergeometric function

I am working with the function $$ _2F_3\left(n+\frac{1}{2},n+\frac{3}{2};n,n+\frac{5}{2},2 n+1;x^2\right).$$ Evaluating the numerical value of this or plotting it indicates that this is always real ...
gb62442's user avatar
  • 25
3 votes
5 answers
273 views

How to a function converges or diverges by comparison test?

Question: $\int_{4}^{\infty}f(x)dx$ is an improper integral, where $$f(x)=\frac{3x+8}{x^4+7}$$ For the comparison test we have to find a comparison function, say $g(x)$ and for any $f(x)>g(x)$, if ...
user305532's user avatar
6 votes
1 answer
177 views

Asymptotic sum of set of natural numbers with positive natural density

Let $d(A)$ denote the natural density of $A\subset \mathbb{N}.$ If $0 < \alpha < 1\ $ and $\ d(A) = \alpha,\ $ then is this enough to imply $$ \lim_{ N\to\infty } \left( \frac{ \displaystyle\...
Adam Rubinson's user avatar
1 vote
0 answers
38 views

Convergence in a divergent double sum by substituting analytic continuation values

There is a definition of the Gram series of R(x) where R(x) is a term in the exact formula of the prime counting function defined as $$R(x) = 1 + \sum_{k=1}^{\infty}\frac{(log(x))^k}{k!k\zeta(1+k)} $$ ...
Horus's user avatar
  • 295
2 votes
0 answers
49 views

Why does the Eisenstein series $E_2(z)$ diverge?

We know that, given a lattice $\Lambda$ of the complex plane, the Eisenstein series $E_{2k} := \displaystyle\sum_{\substack{\omega \in \Lambda \\ \omega \neq (0,0)}} \frac{1}{|\omega|^{2k}}$ converges ...
Squirrel-Power's user avatar
0 votes
0 answers
29 views

Analytical continuation of $-\frac{t^w}{\ln(2)}\sum_{k=-\infty}^\infty \frac{t^{w k}}{e^{\pi i w(1+2k)}-1}$

I'm interested in obtaining the analytical continuation of the following function. Let $w = \frac{2 \pi i}{\ln(2)}$, then let $$F(t) = -\frac{t^{w/2}}{\ln(2)}\sum_{k=-\infty}^\infty \frac{t^{w k}}{e^{\...
Caleb Briggs's user avatar
  • 1,063
3 votes
0 answers
54 views

Let $A \subset \mathbb{N}^{\ast}$. Exhibit $(u_n)_{n \geq 1}$ such as $\forall q \in \mathbb{N}^{\ast}$, $\sum_{n\geq 1} u_n^q$ diverges iff $q \in A$

I am having difficulties with the following exercise on numerical series : Let $A \subset \mathbb{N}^{\ast}$. Exhibit a sequence $(u_n)_{n \geq 1} \in \mathbb{C}^{\mathbb{N}^{\ast}}$ such as $\forall ...
Arthur Filippi's user avatar
1 vote
4 answers
173 views

Asymptotic behaviour of $\displaystyle\sum_{n \leq x} \frac{\log(n)}{n}$

This is my first question, I hope I won't make too many mistakes. I have been given this question as an exercise, but I am struggling to find the solution. I have to calculate the asymptotic behaviour ...
Fralibe's user avatar
  • 21
3 votes
2 answers
131 views

Regularizing infinite sum over $\sqrt{n^2+a^2}$

I am aware that one can use zeta function regularization to obtain \begin{equation} \sum_{n\in \mathbb{N}}n = -\frac{1}{12} \end{equation} I am wondering if it is possible to regularize a similar sum, ...
Kaixiang Su's user avatar
8 votes
2 answers
830 views

Does a recursive definition of a variable make sense at all?

Let $A:=\sum_{n=0}^\infty 2^n$. I was given the following equation: $$A=\sum_{n=0}^\infty 2^n=\sum_{n=1}^\infty2^{n-1}=\sum_{n=1}^\infty(2^n\cdot\frac{1}{2})=\frac{1}{2}\sum_{n=1}^\infty2^n=\frac{1}{2}...
David Krell's user avatar
1 vote
3 answers
119 views

Does the series $\sum_{n=0}^{\infty} \int_0^1 \cos(nt^2) dt$ converge or diverge?

I know we can not write the primitive function of $\cos(nt^2)$ for any $n \in \mathbb{N}$, thus I could not calculate the term $ \int_0^1 \cos(nt^2) dt$, could someone help me? Thank you in advance.
corsair's user avatar
  • 105
0 votes
2 answers
110 views

Why does $\sum_{n=2}^ \infty \frac 1 {n \sqrt {\log n}}$ diverge? [duplicate]

I'm aware that this diverges, but I'm struggling to prove it only using a comparison, or a ratio/root test. Any tips would be greatly appreciated - this isn't even an assignment question, just one ...
errgg45654's user avatar
0 votes
1 answer
43 views

When using the nth term test on a Alternating Series will it always diverge?

If you have a alternating series you and put it though the nth term test you split the limit into two parts multiplying each other. One containing (-1)^n and another containing what its multiplied by. ...
Golden Boy's user avatar
1 vote
3 answers
65 views

Prove that $u_{n} = \frac{1}{n(10n+1)}$ is convergent

Question Prove that $u_{n} = \frac{1}{n(10n+1)}$ is convergent Attempt $u_{n} = \frac{1}{n(10n+1)}$ $u_{n} = \frac{1}{n(10n+1)} = \frac{A}{n}+\frac{b}{10n+1}=\frac{1}{n}-\frac{1}{10n+1}$ $S_{n} = u_{...
lokey8573's user avatar
  • 123
3 votes
0 answers
152 views

A continuous function whose Fourier series diverges at $0$?

I read this at many places such as wiki or elsewhere quote : It is possible to give explicit examples of a continuous function whose Fourier series diverges at $0$. For instance, the even and $2π$-...
mick's user avatar
  • 15.3k
0 votes
1 answer
62 views

Proving convergence theorem on power series

When proving convergence of power series, one encounters the following term: $$\left( 1 + \delta \right)\frac{\sqrt[k]{|a_k|}}{\lim\sup_{n\to\infty}\sqrt[n]{|a_n|}}$$ where $\delta>0$, and the ...
atapaka's user avatar
  • 467
4 votes
1 answer
158 views

Manipulating divergent series for practical applications

I have a series summation of the form $$ \tag{1} S(x) = \sum_{i = -\infty}^{\infty} (-1)^{i}\left[\Phi\left(2ix\right) - \Phi\left((2i-2)x\right)\right], $$ where $\Phi(.)$ is the standard normal ...
ck1987pd's user avatar
  • 1,070
10 votes
2 answers
234 views

If $\{x_n\}$ is positive, decreasing and $\sum x_n=\infty,$ then $\sum x_ne^{-\frac{x_{n}}{x_{n+1}}}=\infty$

The problem is Suppose that $\{x_n\}$ is positive, monotonic decreasing and $\sum_{n=1}^\infty x_n=+\infty$. Prove that $$\sum_{n=1}^\infty x_ne^{-\frac{x_{n}}{x_{n+1}}}=+\infty.$$ This is a past ...
ling's user avatar
  • 1,557
2 votes
2 answers
89 views

Prove that a series is divergent, with $\epsilon,N$

I have this question: Prove that $a_n=\frac{(-1)^nn+1}{n+2}$ is divergent, without proving it by contradiction, or by using any theorems from the book. Prove it by using $\epsilon$ and $N$ notation. ...
therealcain's user avatar

1
2 3 4 5
34