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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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 $k=1$,
$$ \sum_{i=0}^n \left (...
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Determine $\sum_{n=1}^\infty {1/S_n}$ where $S_n=\sum_ {k=1}^n {1/\ln(k+1)}$
Let $S_n=\sum_{k=1}^n {1/\ln(k+1)};$ then $\sum_{n=1}^{\infty} {1/S_n}$
diverges?
In fact, I used the Stolz-Cesaro theorem to find out that it diverges, but I do not completely understand the theorem ...
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How can I show that the given series diverges?
How can I show that this series diverges?
$$
\sum_{n=1}^{\infty} \frac{\bigl(2 + (-1)^n\bigr)^n}{\sqrt{n}\,3^n}
$$
The standard tests kinda fail in this case. I am not getting any clue on how to ...
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Asymptotic evaluation of a series
I am reading a paper in which the authors derive the asymptotics for the following series:
$$
\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(...
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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 $\...
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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>...
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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 ...
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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 ...
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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 $...
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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
$$\sum_{n=1}^{\...
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How to evaluate $\sum \frac{1}{2^{n-r}}\frac{1}{n}$? [closed]
$$\sum_{n=r+1}^{\infty}\frac{1}{2^{n-r}}\frac{1}{n}$$
,where r is a positive integer.
First, I wanna know if this is convergent(I guess so) and WHY.
And if so, how to evaluate it?
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Concoct an alternating series which diverges that is alternating, the limit goes to 0, but is NOT always decreasing. [duplicate]
Basically need an example of a divergent series that is alternating and the limit goes to 0.
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Divergence of Mehler's Hermite polynomial series
According to Mehler's formula,
$$
\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}\...
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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
$$
f(x) = \frac{1}{1-(-x-...
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Question about convergence and divergence of positive series,which means there exists no slowest convergent series.
If positive series $\sum_{n=1}^{\infty} a_n$ is divergent,try to prove exist divergence
positive series $\sum_{n=1}^{\infty} b_n$,which satisfies
$\lim _{n \rightarrow \infty} \frac{b_n}{a_n}=0$
If ...
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T\F: if $\lim_{k\to\infty}\sum_{i=n_k}^{n_{k+1}-1}|\alpha_i|= 0$ then $\sum \alpha_k$ converges.
Let $\sum \alpha_k$ be a bounded series of reals.
Suppose $\{n_k\}_{k=1}^{\infty}$ is a strictly-increasing monotonic sequence of naturals s.t $lim_{k\to\infty} (n_{k+1}-n_{k})= \infty$.
Prove or ...
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If $n\in\mathbb N$, then $\sum_{k=1}^{\infty} \frac{1}{k^n}$ diverges. What's wrong in my proof?
If $n\in\mathbb N$, then $\sum_{k=1}^{\infty} \frac{1}{k^n}$ diverges.
Clearly this is false in general and only holds for $n=1$ . But here's my wrong proof in which I can't find what's wrong.
Let $$...
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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 ...
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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$\frac{\left[2+(-1)^n\right]^n}{n}\left(\frac{1}{3}\right)^n$
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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 ...
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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{...
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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^...
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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)...
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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^{\...
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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 ...
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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} \...
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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 ...
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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 ...
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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,...
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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$ ...
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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 ...
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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).$$
Evaluating the numerical value of this or plotting it indicates that this is always real ...
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How to a function converges or diverges by comparison test?
Question: $\int_{4}^{\infty}f(x)dx$ is an improper integral, where $$f(x)=\frac{3x+8}{x^4+7}$$
For the comparison test we have to find a comparison function, say $g(x)$ and
for any $f(x)>g(x)$, if ...
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Asymptotic sum of set of natural numbers with positive natural density
Let $d(A)$ denote the natural density of $A\subset \mathbb{N}.$
If $0 < \alpha < 1\ $ and $\ d(A) = \alpha,\ $ then is this enough to imply
$$ \lim_{ N\to\infty } \left( \frac{ \displaystyle\...
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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
$$R(x) = 1 + \sum_{k=1}^{\infty}\frac{(log(x))^k}{k!k\zeta(1+k)} $$
...
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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 ...
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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
$$F(t) = -\frac{t^{w/2}}{\ln(2)}\sum_{k=-\infty}^\infty \frac{t^{w k}}{e^{\...
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Let $A \subset \mathbb{N}^{\ast}$. Exhibit $(u_n)_{n \geq 1}$ such as $\forall q \in \mathbb{N}^{\ast}$, $\sum_{n\geq 1} u_n^q$ diverges iff $q \in A$
I am having difficulties with the following exercise on numerical series :
Let $A \subset \mathbb{N}^{\ast}$. Exhibit a sequence $(u_n)_{n \geq 1} \in \mathbb{C}^{\mathbb{N}^{\ast}}$ such as $\forall ...
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Asymptotic behaviour of $\displaystyle\sum_{n \leq x} \frac{\log(n)}{n}$
This is my first question, I hope I won't make too many mistakes.
I have been given this question as an exercise, but I am struggling to find the solution. I have to calculate the asymptotic behaviour ...
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Regularizing infinite sum over $\sqrt{n^2+a^2}$
I am aware that one can use zeta function regularization to obtain
\begin{equation}
\sum_{n\in \mathbb{N}}n = -\frac{1}{12}
\end{equation}
I am wondering if it is possible to regularize a similar sum, ...
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Does a recursive definition of a variable make sense at all?
Let $A:=\sum_{n=0}^\infty 2^n$. I was given the following equation:
$$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}...
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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.
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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 ...
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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. ...
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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_{...
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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 $2π$-...
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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 ...
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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 ...
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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 ...
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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.
...
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