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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17 views

What is the proof of the theorem that the series of reciprocals of square-free numbers diverges?

I am not a mathematician. I am reading David Applebaum's book "Limits, limits everywhere ", in which he gives a proof of the divergence of the series of square-free numbers on page 91. I don't ...
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25 views

When is a series of matrices divergent. How to define divergence in this case?

In Quantum Mechanics we deal with series of operators represented as matrices like $$e^A = 1+ A + \frac{A^2}{2} + \dots$$ and similarly for $\sin(A) $, etc., where $A$ is a matrix. Now my question ...
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14 views

Investigate convergence of a series using comparison test

The series with $a_n$: $a_n = (n^{1/3}-(n-1)^{1/3})/n^{1/2} $ I tried comparing it to the $1/n^2$ and $1/n^{3/2}$ because those definitely converge, but proving the inequality gives rise to pretty ...
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43 views

Determine the convergence of the series $\sum_{n=1} ^{\infty} \frac{5^{n}-2^{n}}{7^{n}-6^{n}}$

Does $$\sum_{n=1} ^{\infty} \frac{5^{n}-2^{n}}{7^{n}-6^{n}}$$ converge? I tried the ratio test but I failed.
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31 views

Convergence of a series involving $e^{in\phi}$

Consider the following series: $\displaystyle\sum_{n=1}^{\infty} \frac{(n+1)e^{in\phi}}{n^2}, \phi \in \mathbb{R}, \phi ≠ 2\pi k$ for $k \in \mathbb{Z}$ Then: $\displaystyle\sum_{n=1}^{\infty} \...
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4answers
54 views

absolute convergent, conditionally convergent or divergent?

I have to find out if $\displaystyle\sum_{n=2}^{\infty}$$\dfrac{\cos(\frac{\pi n}{2}) }{\sqrt n \log(n) }$ is absolute convergent, conditional convergent or divergent. I think it's divergent while the ...
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1answer
46 views

Generalisation of $ \sum \frac {1}{k}-\ln n=\gamma$ to $0 \lt\alpha \lt1 , \sum \frac{1}{k^\alpha}-f(n)= \beta$

looking at Find the value of : $\lim\limits_{n\rightarrow\infty}\left({2\sqrt n}-\sum\limits_{k=1}^n\frac{1}{\sqrt k}\right)$ and knowing that for $$\alpha=1 ,\lim_{ n \to \infty} \sum_{k=1}^n \...
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1answer
51 views

Find the values of $\theta$ for which the series is convergent

How to find the values of $\theta$ for which the series $$\sum_{n=1} ^{\infty} \frac{(1+ \frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}+ ...+ \frac{1}{n})}{n} \cos n\theta$$ is convergent? What I could show ...
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31 views

Show that $\sum_{n=1}^{\infty}\left(\frac{1}{(n+1)!}\prod_{k=1}^nf(k)\right)$ diverges.

Let $f:\mathbb{N}\setminus{\{0}\}\to\mathbb{N}\setminus{\{0}\}$ be an injective function. Show that $$\sum_{n=1}^{\infty}\left(\frac{1}{(n+1)!}\prod_{k=1}^nf(k)\right)$$ diverges. In the quotient ...
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38 views

“Sum” of probabilities equals infinity

We have some independent trials with a certain chance of succeeding, say the $t$th trial succeeds with probability $p(t)$, and fails with probability $1-p(t)$. My question is, if the sum of ...
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48 views

Grandi's series converges to 1/2, proof check

Today, I coincidentally found this. Is this valid ? Why is this happening ? Observe $\int_0^\infty \frac{t^ne^{-t}}{n!} dt = 1$, $\forall n\geq 0$. Let's start $\frac12 = \int_0^\infty e^{-2t} dt=\...
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29 views

Does $(a_n)_{n=1}^{\infty}=\frac{n^{2} \sin\left ( \frac{1}{n} \right )}{2n+1}$ diverges or converges?

$$(a_n)_{n=1}^{\infty}=\frac{n^{2} \sin\left ( \frac{1}{n} \right )}{2n+1}$$ Determine if the following sequence converges or diverges? İf converges, find its value.
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How can an infinite series have two or more values? Which result can be considered “correct”?

Many mathematicians,including Euler and Ramanujan, have assigned the value of the series $1+2+3+4+....$ to be $-\frac{1}{12}$ But in a video by the YouTube channel 'blackpenredpen', it manipulated ...
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26 views

infinite series: which test? [closed]

so, we recently started to learn about series, i feel like i'm confused a little and, i seem like i don't know how to start with this,i guess maybe it's either comparison test, or limit comparison ...
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26 views

Series containing irrational numbers

How can I find if the series $$\sum a_k =\sum (k-\sqrt{k^2+4k+3})$$ is convergent or divergent? Let $a_{k}=k-\sqrt{k^2+4k+3}$. Then $$\lim_{k \rightarrow \infty}\bigg(k-\sqrt{k^2+4k+3}\bigg)\cdot \...
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How to calculate $\sum_{k=1}^n k*2^{n-k}$ [duplicate]

Is it possible to calculate this sum in such a way that it only depends on n? $\sum_{k=1}^n k*2^{n-k}$
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101 views

Showing that $\sum_{n=1}^\infty n^{-1}\left(1+\frac{1}{2}+…\frac{1}{n}\right)^{-1}$ is divergent

How to show that $\sum_{n=1}^\infty \frac{1}{n\left(1+\frac{1}{2}+...\frac{1}{n}\right)}$ diverges? I used Ratio test for this problem and this is the result: $$\lim_{n\to\infty} \left(1+\frac{1}{n}\...
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45 views

the convergence and divergence of a series

If $a_n\ge0$,$S_n=a_1+a_2+\cdots+a_n$, $\sum_{n=1}^{\infty} a_n$ is divergent. I define the following series $$\sum_{n=1}^{\infty} \frac{a_n}{{S_n}^\alpha},\;\alpha\ge 0.$$ I want to know for which ...
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69 views

A geometric series that converges to $\pi$? [closed]

Can someone help me come up with an infinite series (in the sigma notation) for the following scenario: a geometric series $\sum_{n=1}^\infty ar^{n-1}$ that converges to $\pi$
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77 views

Is $\sum_{n=1}^{\infty}(-1)^nn!2^{-n}$ a divergent series?

Determine whether or not the series converge: $\sum_{n=1}^{\infty}\frac{(-1)^nn!}{2^n}$ I did do the ratio test to determine that it diverges but is there a way to not use this method as it isn't ...
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243 views

Conditional Convergence of a Series $\sum_{n=1}^{\infty}\frac{\sin(n)\sin(n^2)}{n}$

I wanted to show that the series $\sum_{n=1}^{\infty}\frac{\sin(n)\sin(n^2)}{n}$ is absolutely convergent or not. My claim is it is conditionally convergent, to show that I used Dirichlet test and ...
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Finding the Sum of series $S = \sum_{n=1}^{243} \frac{1}{n^{4/5}} $.

If $S = \sum_{n=1}^{243} \frac{1}{n^{4/5}} $. Find the value of $\lfloor S \rfloor$ where $\lfloor \cdot \rfloor$ represents the greatest integer function. By approximation using definite integral, ...
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37 views

Convergence of these series: $\sum_{n=1}^{\infty}\frac{a^n}{(1+a)(1+a^2)\dots (1+a^n)}x^n$ , $\sum_{n=1}^{\infty}\frac{2^n+3^n}{n}x^n$

For the first series I applied the criterion of the report and discussed after A. Arriving at the fact that the limit of the report is | a |. When I replaced x with $\frac{-1}{a}$, respectively $\frac{...
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48 views

Geometric Series in Sigma Notation with natural log

So I am trying to solve for this problem below, I attach a screenshot of it, but I just can't seem to find the answer. Is anyone able to solve this? (Extra note: Un = Ur) So for (a) I got 2(1-2-10), ...
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50 views

Convergence of this series: $\sum_{n=1}^{\infty}\frac{\sin(n)\sin(n^2)}{\sqrt{2n^2+1}}(x-1)^{2n}$? [closed]

I have been trying for a long time to find an approach to this problem! I can't figure out how I could determine the convergence of this series!! I tried to determine the radius of convergence with ...
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77 views

Find the value of $a_0a_1a_2\cdots a_n\left(\frac{1}{a_0}+\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}\right)$

Given that the sequence $\left\{a_n\right\}$ satisfies $a_0 \ne 0,1$ and $$a_{n+1}=1-a_n(1-a_n)$$ $$a_1=1-a_0$$ Find the value of $$a_0a_1a_2\cdots a_n\left(\frac{1}{a_0}+\frac{1}{a_1}+\frac{1}{a_2}+...
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37 views

Does $\sum_{k=0}^{\infty}\frac{1}{(k+1)!}\left(\prod_{l=0}^{k}\left(1-ls\right)\right) x^{k+1}$ converge? [closed]

Does $$\sum_{k=0}^{\infty}\frac{1}{(k+1)!}\left(\prod_{l=0}^{k}\left(1-ls\right)\right) x^{k+1}$$ converge for $s\in(0,1)$? If so, to what value?
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52 views

$\sum_{n=1}^\infty b_n x^n$ vs. $\sum_{n=1}^\infty |b_n| x^n$

Let $b_n$ be a complex sequence that is absolutely bounded for large $n$ and $x>0$ such that $$\lim_{x\rightarrow 1} \left| \sum_{n=1}^\infty b_nx^n \right| = \infty \, .$$ Can anyone construct a ...
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29 views

Speed of divergence (connected to harmonic and geometric series)

By monotone convergence (or dominated convergence), let $0< s <1$ we find that as $ s \uparrow 1$ $$ \sum_{k =1 }^\infty \frac{s^k}{k} \to \sum_{k = 1 }^\infty \frac{1}{k} =\infty.$$ My ...
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508 views

Proof of some Riemann problem [closed]

I was wondering if anyone is aware of this proof about the Riemann Hypothesis. https://arxiv.org/pdf/1004.4143.pdf What I like about it is his analytic approach using the behaviour of general series ...
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Proof for Limitation Theorem for Euler-Summation

I'm currently stuck in Hardy's Book 'Divergent Series' at a very small Proof which i can't seem to grasp. Here, $(E,q)$ Summation is defined over $\lim_{m \rightarrow \infty} A_m^{(q)}$ or $\sum_{n=0}...
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40 views

Convergence or divergence of alternating series $\sum _{n=1}^{\infty }\:\frac{\left(-1\right)^nn^2+3}{n\sqrt{9n^2+2}}$

Trying to figure out this problem. Alternating series test doesn't seem to work because finding the limit seems to be an endless loop of L'Hopital's rule in the denominator. Ratio test is inconclusive ...
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77 views

Determining whether the series $\sum_{n=1}^{\infty} \dfrac{2^n+3}{n^{2}+1}$ converges or not

Using either the Ratio Test or the Root Test, I want to know whether the following series converges or not $$\sum_{n=1}^{\infty} \dfrac{2^n+3}{n^{2}+1}$$ I don't think it would be possible to use the ...
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15 views

Convergence of Dirichlet L-series and log functin

I am proving Dirichlet theorem of primes in arithmetic series, and i have a question on the proving convergence of the L series. $\text{log}L(s,\chi)$ is bounded as $s \to 1^+$ and i want to prove ...
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62 views

$\sum_i \sin x_i$ diverges implies $\sum_i x_i$ diverges? [closed]

If the series $\sum_i \sin x_i$ diverges does $\sum_i x_i$ necessarily diverge? I feel like this should be a simple series comparison test or something, but I cannot figure a proof out. If you could ...
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57 views

Series sum Diverge or Converge??

\begin{equation} \sum_{n=1}^{\infty}\frac{n-1}{(n+2)(n+3)} \end{equation} The answer was Divergent, but I solved it as Convergent by using limit comparison test. plz help me solving this question.
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1answer
23 views

Convergence of exponential series

Let $c_j \to \infty$ be a positive sequence. Suppose there exists $s_0 \in \mathbb{R}$ such that $\sum_{j=1}^{\infty} e^{-s_0 c_j} < \infty$. I want to show that the series $\sum_{j=1}^{\infty} e^{-...
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21 views

Limit comparison test converging to a limit

The limit comparison test takes two different series $\sum_{i=1}^{n}a_{n}$ and $\sum_{i=1}^{n}b_{n},$ each with positive terms, then checks $\frac{a_{n}}{b_{n}} \to \ell,$ as $n \to \infty.$ Moreover, ...
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73 views

Convergence of $\sum_{n=2}^{\infty}\frac{1}{n\ln (n)}$ WITHOUT using the integral test.

Yesterday I was tutoring a high school student on the convergence of infinite series. We encounter the following series on his practice assignment: $$\sum_{n=2}^{\infty}\frac{1}{n\ln (n)}$$ He's at ...
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208 views

Evaluating $\int\limits_0^\infty x\operatorname{sech}^3x\ln(\operatorname{sech}x)\ dx$

How to prove that $$I=\int_0^\infty x \operatorname{sech}^3x\ln{(\operatorname{sech}x)}\ dx=\frac{\pi^3}{32}+\frac{\pi}{8}\ln^22+\frac14(3+2G)-2\ \operatorname{Im}\operatorname{Li}_3(1+i)\ ?$$ ...
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Applications of divergent series

There are certain evaluation methods like Cesaro, Abel, and Ramanujan summation, and other techniques like analytic continuation of the zeta function that can be used to assign values to divergent ...
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92 views

Why does 1 + 2 + 4 + 8 + … converge to -1 in the 2-adic number system?

From this page, http://mathworld.wolfram.com/p-adicNumber.html, the norm of $x = \frac{p^a r}{s}$ for r, s relatively prime to p, and a maximized, over the p-adic numbers, is $p^{-a}$. If we take the ...
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164 views

How to construct a closed form formula for a recursive sequence?

In the Wikipedia page of the Fibonacci sequence, I found the following statement: Like every sequence defined by a linear recurrence with linear coefficients, the Fibonacci numbers have a closed ...
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1answer
114 views

How to prove $\sum _{k=1}^{\infty } (-1)^k H_{\frac{2 k}{3}} = -\frac{\pi }{2 \sqrt{3}}+\frac{3 \pi }{8}-\frac{3}{4} \log (2)$?

I stumbled on this problem in the wake of the discussion https://math.stackexchange.com/a/3553902/198592 Can you make sense of the equation in the question involving the harmonic number with a ...
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2answers
25 views

Prime reciprocals series proof

I am currently struggling with the proof of the prime reciprocals series divergence. I've already proved that : $$\prod_{k=1}^n \frac{1}{1-1/p_k} \longrightarrow +\infty$$ Let $n$ be an integer. How ...
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3answers
35 views

Series convergency

I have to prove that this series is convergent: $$\sum_{i=1}^\infty \frac{\sqrt {n^2+1} -1 }{\sqrt[3]n}$$ I try to estimate, that $$\ \frac{\sqrt {n^2+1} -1 }{\sqrt[3]n}~~is ~similar~to ~ \frac{1}{...
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56 views

Convergence of the Average of Partial Sums

The Maclaurin series of $\frac{1}{x+1}$ is of course $\sum_{n=0}^{\infty}\left(-1\right)^{n}\left(x\right)^{n}$, and its interval of convergence is only considered to be $-1<x<1$. Though, it ...
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5answers
90 views

Test divergent series using a comparison test

Test series for convergence / divergence using a comparison test: $$\sum_{n=1}^\infty\frac{n^2+1}{n^3+2}$$ Now, If it would be $$\sum_{n=1}^\infty\frac{n^2+1}{n^3-2}$$ then I could compare it as ...
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1answer
28 views

Need an example of $C^\infty_0$ functions with some conditions

I am learning the distribution theory currently. My professor in class said that, if we defined $J_1(\varphi):=\sum_{n=0}^\infty \varphi^{(n)}(0)$ and $J_2(\varphi):=\sum_{n=0}^\infty \varphi^{(n)}(...
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2answers
34 views

Convergence of double sum implies convergence of inner sums

Suppose $a_i,b_{ij}\in\mathbb{R}$ and $$ 0\le \lim_{n\to\infty} \sum_{i=1}^n\sum_{j=1}^n c_{ij}a_ia_j< \infty. $$ Does it follow that $\sum_{j=1}^\infty c_{ij}a_j$ converges for all $i$? (...

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