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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1answer
88 views

Divergent products.

My question are about divergent products. I'm a Dutch student so i may lack the skil to write it down in the correct notation and forgive my spelling errors. A thing i've found on the internet was ...
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0answers
21 views

If $S_n=\sum_{k=1} ^n{}a_k \to \infty$ as $n\to \infty$ then $\sum_{k=1}^{\infty}{a_k\over {1+a_k}}$ diverges. [duplicate]

Let $a_n>0$ and let $S_n=\sum_{k=1} ^n{}a_k \to \infty$ as $n\to \infty$. Prove $\sum_{k=1}^{\infty}{a_k\over {1+a_k}}$ diverges. I am confused by this sort of sequence\sum thing. How can I use ...
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2answers
25 views

Determine if the series diverge or converge?

So I was wondering what is the best TEST(divergent test, alternating series,power series,ratio test or root test) that we can use for the following series :
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46 views
+50

Expected time until beating an initial try

Consider the following problem: Let $X,X_1,X_2,...$ be i.i.d. random variables. We execute the following experiment. One samples $X$. Then, one samples $X_1$,$X_2$ and so on until the first time the ...
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2answers
61 views

Question about series convergence $\sum_{n=1}^\infty \frac{1}{n}$ and $\sum_{n=1}^\infty \frac{1}{n^2}$

So I have been playing around with convergent series recently and I still have a hard time understanding why $\sum_{n=1}^\infty \frac{1}{n}$ diverges and $\sum_{n=1}^\infty \frac{1}{n^2}$ converges. ...
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0answers
33 views

Closed form of a series with sinh

Is there a simple form for following function (where $a$ and $b$ are constants)? Can it be simplified to a simple form if $a>>b$? $$ u(x) = \sum _{n=0}^{\infty } \frac{ \, (-1)^n ...
2
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1answer
76 views

Is this series: $\sum_{n=1}^{\infty}{{1 \over n} \cos{(n)} \sin{(nx)}}$ convergent?

How can I show that the following series is convergent or divergent ? $$\sum_{n=1}^{\infty}{{1 \over n} \cos{(n)} \sin{(nx)}},x\in \mathbb{R}$$ I want to use Abel-Dirichlet criteria. I've ...
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2answers
80 views

Determine if the following series is convergent or divergent $\sum_{n=1}^{\infty}\frac{ \sin{nx}}{n}$

I've been doing exercises with series for some time and now I've got an exercise that I'm supposed to solve using Abel or Dirichlet criteria. The problem is : Determine if the following series ...
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2answers
27 views

Converges or diverges: $\sum_{n=1}^{\infty}\left [\arctan{\frac{(-1)^n}{n}}+{(-1)^n\over n^2}\right]$

How can I show that the following series converges or diverges ? $$\sum_{n=1}^{\infty}\left [\arctan{\frac{(-1)^n}{n}}+{(-1)^n\over n^2}\right]$$ $\sum_{n=1}^{\infty}\left ...
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2answers
37 views

Determine if it converges or diverges : $\sum_{n=1}^{\infty} \frac {2^n \cdot n!}{1\cdot2\cdots (2n-1)}\cdot \frac{1}{\sqrt{2n+1}}$

Here's the series: $$\sum_{n=1}^\infty \frac {2^n \cdot n!}{1\cdot2\cdots (2n-1)}\cdot \frac{1}{\sqrt{2n+1}}$$ Does it converge or diverge ? Thanks
3
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0answers
121 views

Question concerning the divergence of a kind of “hyperharmonic” series different than the definition of Conway and Guy

Conway and Guy defined $$H_k^0=\sum_{n=1}^k\dfrac1n$$ and $$H_k^r=\sum_{n=1}^kH_n^{r-1}$$ for $k,r\in\Bbb Z^+$. I would prefer a definition of an $r$-hyperharmonic number to have some chance of ...
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0answers
30 views

Explicit analytic continuation of the zeta function and shift operators

Is there a way to compute the radius of convergence of the expansion of the zeta function, e.g. around $a=2+2i$? We have $\zeta(a)=\sum_{n=1}^\infty n^{-a}\approx 0.867.. - i\,0.275.. $, but I ...
2
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2answers
84 views

$\prod\limits_{k = 0}^\infty {(1 - {p_k})} = 0$ if and only if $\sum\limits_{k = 0}^\infty {{p_k}} $ diverges

Let $p_0,p_1,p_2,...$ be real numbers in $(0,1)$. I am trying to prove that $\prod\limits_{k = 0}^\infty {(1 - {p_k})} = 0$ if and only if the series $\sum\limits_{k = 0}^\infty {{p_k}} $ ...
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3answers
47 views

Can I use the index of a series for help with divergence?

I was studying this series: $$\sum_{n=2}^{\infty}\dfrac{5}{7n+28}$$ I know that it's an increasing, monotone sequence. Also, I know I can rewrite as: $$\sum_{n=2}^{\infty}\dfrac{5}{7(n+4)} = ...
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0answers
14 views

trouble with convergent and divergent series

Determine whether zn=nth root of(e^n^2(i-1)) is convergent or divergent? i have having trouble with this. How to proceed with this?
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2answers
72 views

convergence of $\sum_{n=1}^{\infty}(\sqrt{n^2+7}-\sqrt[3]{n^3+8n+1})\ln(1+1/n)$

$$\sum_{n=1}^{\infty}(\sqrt{n^2+7}-\sqrt[3]{n^3+8n+1})\ln(1+1/n)$$ I eventually reached $\sum(n(\sqrt{1+7/n^2}-\sqrt[3]{1+8/n^2+1/n^3})\ln(1+1/n))$ and I think this is a dead end. I have no other ...
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2answers
37 views

check the convergence of the series

So I have the following series: $$\sum_{n=2}^{\infty}\frac{e^n}{e^{n\sqrt[n]{n}}(ln(n))^2}$$ I thought that this series does not converge. $\sqrt[n]{n}\rightarrow 1$ so I thought that for large $n$ ...
4
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5answers
229 views

$\sum_{n=0}^{\infty} \frac{1}{2n+1} = 0.66215 + \frac{1}{2}\log(\infty)^{3}$

Just finished Euler: The Master of us All. A good fraction of the book is dedicated to explaining in why certain divergent series were useful in proving Euler's theorems, but this one is never ...
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1answer
31 views

Convergence of series for specific values of $\lambda$.

Let $\lambda$ be a positive real number. For which values of $\lambda$ does the following series converge? $$\sum_{n=1}^\infty \frac{n^{-\lambda}}{1+\lambda^{-n}}$$ I can see that the series ...
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17 views

Divergence proof problem in introductory analysis text.

The problem is this: Show that if $a_n > 0$ and $\lim_{n\to \infty} na_n = L$ with $L \neq 0$, then the series $\sum a_n$ diverges. (from Abbott's Understanding Analysis, p. 68). I want ...
4
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1answer
61 views

Series convergence - factorial over products

Does $$ \sum_{n\ =\ 1}{n!\over \left(\,\sqrt{\,2\,}\, + 1\,\right) \left(\,\sqrt{\,2\,}\, + 2\,\right)\ldots \left(\,\sqrt{\,2\,}\, + n - 1\,\right)\left(\,\sqrt{\,2\,}\, + n\,\right)}\quad $$ ...
4
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5answers
275 views

Does ${\sum_{n=1}} \log(1+{1\over n})$ diverge or converge?

How do I find out if ${\sum_{n=1}} \log(1+{1\over n})$ diverges or converges? Wolfram recommends me to use comparsion test, but I do not know series which diverges and less than this.
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1answer
34 views

Show that this sum is divergent

How can I argue that this sum i divergent? $$\sum_{j=1}^\infty \left( \frac{1}{j\beta}+\frac{\delta}{(j-1)\beta^2}+ \frac{\delta^2}{(j-2)\beta^3}+ \cdots + \frac{\delta^{j-1} }{\beta^j} \right)$$ ...
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2answers
55 views

Proving divergence of series using a recursive relation

I have been thinking for an hour about this problem but could not find any way to solve it. Let's $0\lt a_n \lt a_{n+1}+a_{n}^2$, prove that $\sum_{n=1}^{\infty}a_n$ is divergent. Any hints and ...
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1answer
27 views

radius of convergence of hypergeometric function

Looking up information on the Bessel function there is a formula as $|z| \to \infty$: $$ I_0(z) \approx \frac{e^z}{\sqrt{2\pi z}} {}_2F_0( \tfrac{1}{2}, \tfrac{1}{2}, \tfrac{1}{2z}) = ...
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0answers
47 views

Asymptotic behavior of divergent $p$-series

I am intertested in the asymptotic scaling behavior of the divergent $p$-series $$ \sum_{k=1}^n \frac{1}{k^p} $$ for $0<p<1$, i.e., is there a closed-form sequence $a_n$ so that $$ \lim_{n \to ...
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0answers
53 views

Convergence of a subseries of the series $\sum_{n \ge 1} \frac{1}{p_n}$, where $p_n$ is the $n$ th prime.

Let $p_n$ be the $n$th prime number. Does the following series $$ \sum_{n \ge 1} \frac{1}{p_{p_n}} = \frac{1}{3} + \frac{1}{5} + \frac{1}{11} + \cdots $$ converge or diverge? Similarly, I am so ...
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1answer
65 views

When do regularization methods for divergent series disagree?

Sometimes, it is possible to take a divergent series (in the sense of its sequence of partial sums failing to converge) and "regularize" it using one of a variety of methods to assign it a meaningful ...
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3answers
43 views

Convergent of a series

For this question, I can successfully prove that the series of |sin(n^0.5)|/n^1.5 converges but I don't understand why the solution can just say " |sin(n^0.5)|/n^1.5 converges, hence the original ...
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2answers
147 views

Is the sum of positive divergent series always divergent?

If two positive terms series $\sum_{n=1}^{\infty} a_n, \sum_{n=1}^{\infty} b_n$ are divergent, $\sum_{n=1}^{\infty} (a_n+b_n)$ is also divergent. I thought is was obvious, but I saw a counterexample ...
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2answers
42 views

Absolute convergence of an infinite series and p-series test

Why does the infinite series $\sum_{n=1}^{\infty}\frac{(-1)^n}{\sqrt n}z^n$ where $z\in \mathbb{C}$ converge absolutely for $|z|<1$. Doesn't the series diverge because if we apply the absolute ...
2
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5answers
104 views

Analysis: Prove divergence of sequence $(n!)^{\frac2n}$

I am trying to prove that the sequence $$a_n = (n!)^{\frac2n}$$ tends to infinity as $ n \to \infty $. I've tried different methods but I haven't really got anywhere. Any solutions/hints?
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1answer
32 views

convergence of the series to inf or not

let $a_n = \dfrac{e^{-(1/2) \times a^2 \times\log(n) }}{a\sqrt{2\pi \log(n)}} $, $a$ is a constant, and the question is if $S_n = \sum a_n$ converge to a finite number. I wonder if I should ...
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2answers
33 views

Divergence Theorem to calculate flux

Take the vector field given by: $F= (y^2+yz)i+(\sin(xz)+z^2)j+z^2k$ a) Calculate the divergence, $\operatorname{div}F$. b) Use the divergence theorem to calculate the flux $$\int_S F\cdot dA $$ ...
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1answer
30 views

Determine the convergence of the following series

$$\sum_{k=0}^\infty {3^{k\ln k} \over {k^k}}$$ I need to determine the convergence of this series. I know it diverges, but how do I prove this?
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23 views

sum of divisors function $\sum \tau(n) = \frac{1}{4}$

These notes on multiplicative number theory mention the convolution $ 1 \ast 1 = \tau$ (where $\tau$ is the divisor function not Ramanujan tau function. Therefore $$ \bigg(\sum \frac{1}{n^s} ...
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33 views

Solutions of fractional linear dynamical systems

The Mittag-Leffler function is defined as: $$ E_\alpha(\tau) = \sum_{k=0}^{\infty}\frac{\tau^k}{\Gamma(\alpha k + 1)}, $$ which can also be defined, analogously, for matrices $A\in\mathbb{R}^{n\times ...
3
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1answer
115 views

Is there a metric in which $1+2+3+4+\cdot$ converges to $-\frac1{12}$?

It is well known that the sum $1+2+3+4+\ldots$, which tends to infinity in the regular sense, can be assigned the value $-\frac{1}{12}$ by different means, e.g., zeta regularization or different ...
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1answer
31 views

Did I correctly verify the convergence of this series?

I want to find if the following series is convergent. $$\sum_{n=1}^\infty \frac{(1+\frac{1}{n})^nn^2-7n}{n^3+3n^2+1} $$ I use the asymptotic criterion for series convergence. $$ ...
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2answers
45 views

Alternating Series Test: example which meets all but the decreasing condition?

Leibniz's Alternating Series Test The series $\sum (-1)^{k-1} u_k$ converges if: $u_k \geq 0$ $u_{k+1} \leq u_k$ $u_k\rightarrow0$ as $k\rightarrow\infty$ I need to find an ...
3
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4answers
69 views

Show that $f(n) = \left(\frac{\sqrt 2}{2}(1+i)\right)^n$ diverges

I am having this sequence: $$f (n)= \left( \frac{\sqrt 2}{2}(1+i)\right)^n$$ I think it is divergent, because I found a subsequence that is divergent: The subsequence $4n$ shows that the sequence ...
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0answers
18 views

Summation methods and entropy

I am aware of the theory of divergent series, but don't know much of it. If you have a text to recommend, I'd be glad to hear it. Suppose I have an infinite-dimensional probability vector $\mathbf{p} ...
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2answers
54 views

$\sum^{\infty}_{n=1}(n-\sqrt n)/(n^{2}+5n)$ diverges

Show that $$\sum^{\infty}_{n=1}\frac{n-\sqrt n}{n^{2}+5n}$$ diverges. I have tried Root test, ratio Test, Cauchy condensation Test but all have failed. I think this has to be done by Comparison Test ...
2
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2answers
80 views

Show that a Series Diverges

Question: Let a sequence ($a_n$) have the property $\lim \limits_{n \to \infty} na_n = a > 0$ Show that the series $\sum_{n=1}^\infty a_n$ diverges Attempts: Basically, I firstly tried ...
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1answer
24 views

Alternating Series Test for Divergence

Alternating Series Test if the alternating series $$\sum^\infty_{n=1}(-1)^{n-1}b_n=b_1-b_2+b_3-b_4+b_5+...\;\;\;\;\;b_n\gt0$$ satisfies $$(\text{i)}\;\;b_{n+1}\leq b_n\;\;\;\;\;\text{for all ...
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4answers
69 views

Sum of all triangle numbers

Does anyone know the sum of all triangle numbers? I.e 1+3+6+10+15+21... I've tried everything, but it might help you if I tell you one useful discovery I've made: I know that the sum of ...
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1answer
219 views

Square root of a divergent series diverges.

If we let $\sum a_n$ converges, we cannot say anything about convergence of $\sum \sqrt{a_n}$ The counter examples being $a_n = \frac{1}{n^2}$ and $a_n = \frac{1}{n^4}$ But what about if the ...
0
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2answers
56 views

$\sum a_n = \sum{(\sqrt{n+1} - \sqrt{n})}$ Prove divergence [duplicate]

$\sum a_n = \sum{(\sqrt{n+1} - \sqrt{n})}$. I want to show that this diverges. I think the only way I can do it, is by using the comparison test, and finding a series less than it that diverges, but I ...
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0answers
75 views

Interpretation of Ramanujan summation of infinite divergent series

I am not mathematician by any means so this question might be rather stupid. I came across this Wikipedia article on Ramanujan's summation and found this bewildering formula, 1 + 2 + 3 + ... = - ...
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0answers
33 views

Resummation of a series from an integral

Let's consider the integral $$\int_0^{\infty}e^{-gx^2-x}dx$$ If I'm not mistaken, $$\int_0^{\infty}e^{-gx^2-x}dx=\frac{\sqrt \pi}{2\sqrt g}e^{1/4g}\left(1-\mathrm{erf}\left(\frac{1}{2\sqrt g}\right) ...