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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Test for the convergence of the series $1/5 + 2!/5^2 + 3!/5^3+\ldots $
The series is
$1/5 + 2!/5^2 + 3!/5^3+\ldots $
It's $n$th term will be $n!/5^n $
The problem I'm facing that I'm getting infinity using D'Alembert's ratio test.
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Series Test, Calculus 2 [closed]
I want to determine whether the series
$$\sum_{n=3}^{\infty} \frac{1}{\ln(\ln(n))}$$
converges or diverges, and I don't know what test I should use.
Thank you.
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Improper integral comparison test of $\int _0^1\frac{\cos\left(x\right)}{x^{1/3}}dx\:$
I'm trying to find the convergence/divergence of this integral but
I can't seem to find an integral to compare to
$$\int _0^1\frac{\cos\left(x\right)}{x^{1/3}}dx$$
I tried to compare to integral of $\...
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Convergence of a series induced from given two series
I could not get any counter example so I am asking this question.
Given $\frac{1}{n}\sum_{k=0}^{n}ka_k\to 0$ and $\frac{1}{n}\sum_{k=0}^{n}kb_k\to 0$, as $n\to \infty$, where $a_k,b_k \in (0,1)$.
Is ...
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Am I correct about this series question?
The problem was as follows: Let $a_n$ be a non-negative series. Assuming $\sum_{n=1}^{\infty}a_n$ converges does
a) $\sum_{n=1}^{\infty}\left(a_n\right)^{\frac{3}{2}}$ converge?
b) $\sum_{n=1}^{\infty}...
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2
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Does this series converge or not?
$$\sum _{k=2}^{\infty }\:\frac{1}{\sqrt{k}\left(\ln k\right)^{\ln k}}$$
I've tried limit comparison with $\frac{1}{\sqrt{n}}$ and $\frac{1}{n^2}$ and also Cauchy condensation test but nothing seems to ...
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How does the divergent sum $\sum_{n=1}^\infty\cos(2n\gamma)\sin(2nt)$ correctly evaluate an integral? Surely distributions don’t apply here
$\newcommand{\d}{\,\mathrm{d}}\newcommand{\res}{\operatorname{Res}}$Note: I don’t know any distribution theory myself, but I was informed by someone else and hinted to by this answer that my problem ...
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Does the series $\sum_{n=1}^{\infty}(n^\frac{1}{n}-1)$ diverge? [duplicate]
I came across the series $\sum_{n=1}^{\infty}(n^\frac{1}{n}-1)$ and I am not able to apply any of the convergence test here. I know that the series $\sum_{n=1}^{\infty}n^\frac{1}{n}$ and $\sum_{n=1}^{\...
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Does the analytic continuation method for infinite series always work and is unique?
Sorry if this is worded poorly, but I don’t know enough in the subject to word it better:
$$\sum_{n=0}^{\infty}n=-1/12$$ according to $$\zeta(-1)=\sum_{n=1}^{\infty}\frac{1}{n^s}=-1/12$$
Is there ...
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Convergence of $\sum_{k=1}^\infty \frac{3k}{k^2+k} \sqrt\frac{\ln(k)}{k}$
I'm having a hard time doing this problem. As this is homework, I'd appreciate a guidance towards a solution, not a full answer. Thanks in advance.
I need to prove that the series
$$\sum_{k=1}^\infty \...
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The Method used in the solution
$$ \sum_{k=1}^∞ \frac{(x^k)} {(k^2)}
$$
The question is, Check if it is divergent.
Solution:
Step1:
$$ R_1 = \frac{1}{lim \frac{k^2}{(k+1)^2}} =1
$$
can someone explain step 1.Which method is used ...
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How can I prove $\displaystyle\sum_{n=1}^{\infty} \frac {(-1)^n \sin^2(8n)}{n}$ diverges?
Again, the series is $$\sum_{n=1}^{\infty} \frac {(-1)^n \sin^2(8n)}{n}.$$
Wolfy says it converges, at first I thought it would converge, but after thought I think it diverges, mathway says it ...
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If $\varepsilon_n \rightarrow 0 $ do we have that $\sum_{n<N} \varepsilon_n b^n = o (\sum_{n<N} b^n)$
If $\varepsilon_n \rightarrow 0$ and $b > 1$ do we have that $\sum_{n<N} \varepsilon_n b^n = o (\sum_{n<N} b^n)$ as $N \rightarrow +\infty$ ?
I've tried an Abel transformation but I was not ...
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Convergence (or not) of $\sum_{n=1}^{\infty}\frac{1!2!\cdots n!}{n^n}$
The title says for itself: Does the series
$\sum_{n=1}^{\infty}\frac{1!2!\cdots n!}{n^n}$ converges? So, as soon as I've saw the factorial, could not help but use the ratio test:
\begin{align*}
\lim \...
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Sum of Reciprocal of Repeated Products [closed]
For what real values of $c\in\mathbb{R}$ does the following series converge and diverge?
$$ \sum_{k=1}^\infty\frac{1}{\prod\limits_{\substack{1\leq j\leq k \\ j\neq -c}}\left(1+\frac{c}{j}\right)} $$
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Does this recurrence relation of exponentials diverge?
Does the following recurrence relation diverge?
$$x_{k+1} = x_k - \frac{e^{-x_k}}{(1+e^{-x_k})^2}$$
where $x_0 <0$.
I have plotted the first one million steps of the sequence using $x_0=-1$ ...
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How to check the convergence or divergence of this alternating series: $\sum_{n=1}^{\infty}\frac{(-1)^n}{1+\sqrt{n}}$?
Check the convergence or divergence of this alternating series: $$\sum_{n=1}^{\infty}\frac{(-1)^n}{1+\sqrt{n}}$$
My attempt:
I know that
$$\frac{1}{\sqrt n}>\frac{1}{n}\tag 1$$
we conclude that $\...
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Does the series $\sum_{n=1}^{\infty}\frac{\sin n}{n}(-1)^{n}$ converges or not?
From other questions we could know the series $\sum_{n=1}^{\infty}\frac{\sin n}{n}$ converges (using the Euler equaiton) and $\sum_{n=1}^{\infty}\frac{|\sin n|}{n}$ diverges (using the Euler–Maclaurin ...
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Error term of $( a_0+a_1x^{-1}+\mathcal O(x^{-2}))(b_0+b_1y^{-1}+\mathcal O(y^{-2}))$
Suppose
$$
f(x)= a_0+a_1x^{-1}+\mathcal O(x^{-2}),\quad x\to\infty
$$
and
$$
g(y)= b_0+b_1y^{-1}+\mathcal O(y^{-2}),\quad y\to\infty.
$$
How would we quote the error term of the product $f(x)g(y)$ for ...
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Absolute convergence and conditional convergence of the sum $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}\cdot(1+\frac{1}{n})^n}{n}$
I am trying to solve an exercise from an old exam from my university which no solution is uploaded.
Basically, I have to determine if the series:
$$\sum_{n=1}^{\infty}\frac{(-1)^{n+1}\cdot(1+\frac{1}{...
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Convergence $\ln\left(1+\frac{1}{n^2}\right)$ [duplicate]
I´m asking for some help, since I am stuck with a proof for convergence like follows:
$$\sum\limits_{n=1}^\infty\ln\left(1+\frac1{n^2}\right)$$
I tried to separate it:
$$\sum\limits_{n=1}^\infty\ln\...
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Interpreting a potential bias of the Mertens function
In this question of some years ago on MO, the presence of a negative bias for the Mertens function was hypothesized.
A key point for such a problem is how the bias is defined. For example, if we focus ...
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Valid proof that Euler's Constant $\gamma$ is between $0$ and $1$?
I was wondering if there is any reasonable way/theory to do calculations with divergent limits of a sequence. I was trying to prove that Euler's constant $$\gamma = \displaystyle{\lim_{n \to \infty}}
\...
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Closed form for the sum of the series $\sum^\infty_{n=1} \left( {(-1)^n}\left( 1+n\ln(\frac{2n-1}{2n+1}) \right) \right)$
If $a>2$ then $\sum\limits^\infty_{n=1} \left( {(-1)^n}\left( 1+n\ln(\frac{an-1}{an+1}) \right) \right)$ diverges by divergent test. Does it converge if $a=2$? Is it possible to find an exact form ...
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Convergence of series with logarithm upon polynomial as terms [duplicate]
Consider the following sequence $a_n=\dfrac{\ln n}{n^p}$ where $p>1$. I want to check for the convergence of the series of that sequence i.e. $\displaystyle \sum_{n=1}^{\infty}a_n$.
I intuitively ...
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Question about convergence of a series.
First of all, Happy new year!
Sorry for the non specific title, There was not enough space for a good description.
I have been trying to find a example for the following function $f$:
Does there ...
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Continuation of an asymptotic series originally defined for $z>0$ to $z<0$
Consider the well-known asymptotic expansion of the $\Gamma%$-function:
$$\Gamma(x)\sim\left(\frac{x}{e}\right)^x \sqrt{\frac{2 \pi }{x}}\left[1+\frac{1}{12 x}+\frac{1}{288 x^2}-\frac{139}{51840 x^3}-\...
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$\sum_{n=1}^{\infty} a_n$ is a series and {$s_n$} is the sequence of partial sums. If $s_n = \frac{n^2+1}{4n^2-3}$, find $\sum_{n=1}^{\infty} a_n$
Only idea I have is to use $a_n = s_n-s_{n-1}$, which would mean $a_n = \frac{n^2 + 1}{4n^2 - 3}-\frac{(n-1)^2 + 1}{4(n-1)^2 -3}$. But I'm not sure if I'm thinking in the right direction. Or would ...
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Does the series $\sum_{n=2}^{\infty}\frac{1}{1+(-1)^{n}\sqrt{n}}$ converge or diverge? [duplicate]
As you've seen from the title, I'm wondering whether the series $\sum_{n=2}^{\infty}\frac{1}{1+(-1)^{n}\sqrt{n}}$ converges or diverges? I'm struggling!
My first thought was to rewrite it like this to ...
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Is there a divergent sequence such that for every n in N it is possible to find n consecutive twos somewhere in the sequence.
Is there a divergent sequence such that for every n in N it is possible to find n consecutive twos somewhere in the sequence.
My thoughts are sequences as : { 1 , 2, 2, 2, 2, 3, 2, 2, 2, 4, ...} or ...
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1-1+1-1+... = k+1/2?
Let $$F(s) = \sum_{n=0}^{\infty} f_{n}(s)$$ be a complex analytical function defined by a series (not necessarily a power series) that absolutely converges on an open set $s \in U$. Assume that the ...
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How to make this series converge?
Let $\lambda>0$ be fixed, and $a,b>0$ positive real numbers. We have a series which is defined as
$$\sum_{n=0}^{\infty}\bigg(\prod_{j=1}^{n}\frac{1}{1+\lambda\frac{1}{a+b(j-1)}}\bigg).$$
Is ...
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Do the elements of this series converge to zero?
Let $\{a_n\}\subset [0,\infty )$. It is well know that if $$\sum_{n \in \mathbb{N}}a_n < \infty$$ then we must have:
$a_n \to 0$ as $n \to \infty$.
Now suppose that $a_n=a_n(u)$ with $\{a_n\}\...
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How do I check whether the sequence converges uniformly?
The sequence is:
$$f_n=x^n-x^{n+1}=x^n(1-x),\quad x\in[0,1]$$
I need to check whether it converges uniformly, but I'm having doubts about the answer for $x\in(0,1)$. Below is what I've tried.
I apply ...
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The sum of the series(in functional series)
The task is to find the sum of the series:
I can calculate the sum of an ordinary positive number series, but as I understand it, this series is not an ordinary positive number series. Tell me how to ...
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Decide whether the sum is convergent or divergent
$$\sum^{\infty}_{n=0}\frac{1}{n!}(\frac{n}{e})^n$$
I have tried both ratio test and root test, both results in $1$, which cannot give any conclusion. Also the series itself goes to zero as $n$ goes to ...
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The harmonic Series sequence.
Well! I was going through harmonic series from mathworld.worlfram I found harmonic numbers are really tough to calculate I was scribbling and wrote this thing
$$\sum_{k = 1}^{x}H_k= \sum_{k =1}^{x}\...
user992847
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Cancelling values in sum of an infinite series
Question: Find the sum of all integer values of c such that $x^2 +cx+\frac{1}{4}c$ has two real, distinct roots.
The discriminant must be greater than zero for a quadratic to have two real, distinct, ...
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Expansion of square root of (a-b)
I am interested in expanding:
$$\sqrt{a-b}$$
Assuming $a>0$ and $b>0$, and that $a>b$.
When I write this in Wolfram alpha, the expansion is as follows:
$$\approx \sqrt{-b} +\frac{a}{2\sqrt{-b}...
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0
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Proof of existence of a rearrangement that diverges to infinity for a conditionally convergent sieries
I came across the Wikipedia page on the Riemann series theorem here (section 4.2) which includes a proof that for a conditionally convergent sequence $a_n$ there exists a rearrangement such that $\...
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Existence of a bijective function from the set of natural numbers to itself such that the following condition satisfy
Does there exist a bijective function $g:\mathbb{N}\to \mathbb{N}$ such that $\sum_{n=1}^{\infty}\frac{g(n)}{n^2}<\infty$
My approach
Suppose it does then the sequence $(\frac{g(n)}{n^2})_{n\in \...
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Show that $\sum_{n=3}^{\infty} \frac{1}{(\log \log n)^{\log \log n}}\quad $ diverges.
This problem comes from Apostol's Ex. 8.15
Show that $\sum_{n=3}^{\infty} \frac{1}{(\log \log n)^{\log \log n}} \;$ diverges.
My attempt:
Note $ \log \log n$ is increasing and positive, so the term ...
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1
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The radius of convergence of $\sum \left(\frac{z^2-1}{z^2+1}\right)^n$
I'm trying to find the radius of convergence of complex series
$$S=\sum_0^\infty\left(\frac{z^2-1}{z^2+1}\right)^n$$
with the help of ratio test.
With simple observation that the term $a_n$ become ...
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1
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Test for Divergence Using Divergence Criterion for Functional Limits
Can I prove divergence of $\sum_{n=1}^\infty\frac{1}{\cos(n)+2}$ by showing that $\lim_{x\to \infty}\frac{1}{\cos(n)+2}$ does not exist? I know there is very simple way to solve it, but I am thinking ...
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Convergence/Divergence of series
I'm stuck in a problem "Using Limit Comparison Test determine for which values of p the series converges or diverges to" $$ \sum_{k=1}^\infty\frac{1}{k^p*\log_{10}k}$$
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Is the series expansion of $e^x$, $\sin x$ & $\cos x$ valid for any real $x$? [closed]
$$e^x=1+\frac{x}{1!}+\frac{x^2}{2!}+\cdots$$
$$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$$
$$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots$$
Are the above valid for any real $x$?
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convergence of a series by permuting its elements [duplicate]
Let $f: \mathbb{N} \to \mathbb{N} $ be a bijective function, the series $ \displaystyle \sum_ {n = 0}^{\infty} a_ {n} $ converges if and only if the series $ \displaystyle \sum_ {n = 0}^{\infty} a_ {...
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Prove that the series $\sum_{n=1}^{\infty} x_n$ diverges in a Hilbert space $~X$
Let $~X~$ be a Hilbert space and $~(e_n)_n~$ be an orthonormal basis for $~X.~$ Let us consider a sequence $~~(x_n)_n~~$ in $~~X~~$ given by
$$x_n=e_n+\frac{e_n}{n+1},~~~~\text{ for all }~n \leq 1.$$
...
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In an alternating series remainder where the 1st term in remainder is a negative, why is the approximate series an overestimate?
Saw this answer but it doesn't go deep enough to help understanding this.
I need help identifying my knowledge gap as I struggle to understand why it is an overestimate. My thinking below results in ...
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4
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Does $\sum_{n=1}^\infty \left(1+\frac{1}{n}\right)^{n^2}\left(\frac{1}{e}\right)^n$ converge?
$$\sum_{n=1}^\infty \left(1+\frac{1}{n}\right)^{n^2}\left(\frac{1}{e}\right)^n$$
With the Root Test I get: $e\times \frac{1}{e}=1$, which doesn't determine whether it converges or not.
With the ...
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