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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0answers
19 views

Requesting help on understanding series [on hold]

Is the tangent of a positive convergent series still positive?
-1
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1answer
32 views

I have a feeling that these statements on my homework are true, but how would I prove it? [on hold]

If $\sum_{n=1}^{\infty}a_n$ and $\sum_{n=1}^{\infty}b_n$ are both absolutely convergent series with all positive terms then $\sum_{n=1}^{\infty}a_n/b_n$ is absolutely convergent. If the power ...
-1
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1answer
40 views

Give an example of a divergent and a convergent series such that the following holds: [closed]

I'm having trouble with this: I need to find an example of a divergent series $\sum_{n=1}^\infty a_n$ of positive numbers $a_n$ such that $lim_{n \rightarrow \infty }$ $a_{n + 1}/a_n$ = $lim_{n ...
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1answer
35 views

Puzzled at this alternating series problem.

I have rechecked this problem so many times, and even my tutor got stuck on this. Can someone tell me what I did wrong? My homework says I got at least one question wrong. And my tutor was confused ...
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0answers
24 views

How can I show that the series below converges/or diverges? [closed]

I don't know how to approach this problem. I would appreciate any ideas/help. [(1/2)*(1/2)]/(9*7*25*1!) +[(1/2)(3/2)(3/2)]/(11*9*49*2!) +[(1/2)(3/2)(5/2)*(5/2)]/(13*11*81*3!) + ...
-1
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4answers
59 views

Root test for $\sum_{n=8}^\infty \left(1+\frac{3}{n}\right)^{n^2}$

I got that it would be infinity and diverge, but my answer seems to be incorrect. What did I do wrong? $$\lim_{n\to\infty}\left(1+\frac3n\right)^{n^2}= \lim_{n\to\infty} ...
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1answer
22 views

Root Test for Convergence or Divergence (ln problem)

I'm stuck at this problem. How would I approach this (I have to use the root test for this one)? The ln and e and my power is throwing me off. This is how far I have gotten.
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1answer
30 views

estimates for Abel's theorem

Suppose $a_1,a_2,\dots$ is a sequence of real numbers with $\displaystyle\sum_{n=1}^\infty a_n =s<\infty$. For $0<z<1$, define $f(z):=\displaystyle\sum_{n=1}^\infty z^n a_n$. By Abel's ...
1
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1answer
55 views

Infinite series $\sum_{n=1}^{\infty}nx^{n+1}$ does not comply to any of my (known) tests

I am attempting to find the interval of convergence for $$\sum_{n=1}^{\infty}nx^{n+1}$$ The lower bound, x = -1, would be tested by determining if $$\sum_{n=1}^{\infty}n(-1)^{n+1}$$ diverges. ...
1
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1answer
39 views

Is $H(\theta) = \sum \limits_{k=1}^{\infty} \frac{1}{k} \cos (2\pi n_k \theta)$ for a given sequence $n_k$ equal a.e. to a continuous function?

I am studying Furstenberg's article Strict ergodicty and transformation of the torus and I'm stuck with the following construction. Define sequence $(v_k)_{k \in \mathbb{N}}$ as $v_1 =1, ...
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2answers
21 views

Use ratio test to test for convergence or divergence

I have online hw and it tells me if my answer is correct or not. It said that my answer for this problem is incorrect: Can someone tell me what I did wrong? Also I might be asking alot of these ...
1
vote
1answer
41 views

Showing convergence and divergence

Say I have: $(x_n)$ a sequence of real numbers such that $\sum x_n$ which converges conditionally and implies $\sum x_{2n}$ diverges. I want to show that $x_{2n}$ does not in general converge. So I ...
1
vote
1answer
38 views

$\sum \limits_{k=1}^{\infty} \frac{1}{k} \cos 2\pi k \theta$ does not converge as $\theta \rightarrow 0?$

We know that the series $H(\theta) := \sum \limits_{k=1}^{\infty} \frac{1}{k} \cos 2\pi k \theta$ is convergent for every $\theta \in (0,1)$ and for $\theta = 0$ the series tends to $+ \infty$. Is it ...
-6
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4answers
101 views

Proof of 1+1+1+1+… = 0 [on hold]

I have thought that 1+1+1+1+... is equal to -1/2. However I found this proof based on the assumption of 1+2+3+4+5+... = -1/12 and -1+(-2)+(-3)+(-4)(-5)+... = +1/12 1+1+1+1+1+1+... is equal to ...
1
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4answers
41 views

$d_n=(1+(2/n))^n$ converge or diverge and find the limit?

I know the answer is $e^2$ and I'd like to use L'Hopital's rule because this is an indeterminate form. Can someone explain how to get there? $$d_n=\left(1+\frac{2}{n}\right)^n$$
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1answer
74 views

Which of the following is true for $\int_{1}^{0} x\ln x\, \text dx$?

Which of the following is true for $\int_{1}^{0} x\ln x\,\text dx$ it is equal to $−1/4$ it is divergent it is equal to an irrational number does not have a closed form it is impossible to ...
3
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2answers
74 views

How did Rudin conclude his argument there is no “boundary” between convergent and divergent series?

I lost my baby Rudin book on real analysis book but I recall a pair of results in homework exercises that he seemed to indicate that there is no "boundary" between convergent and divergent series of ...
0
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1answer
26 views

Convergence behavior of $\sum_p \frac{1}{p \log p}$ and generalization.

The harmonic series $$\sum_{n\in\mathbb N} \frac{1}{n}$$ is well known to be divergent. Using Cauchy condensation test one immediately sees that even $$\sum_{n\in\mathbb N} \frac{1}{n\log n}$$ is ...
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1answer
20 views

Help with series convergence and divergence concept

When you are trying to figure out whether a series converges or diverges why can you test for convergence at any term in the series as opposed to having to start at the beginning of the series. Is it ...
0
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0answers
42 views

What is the sum of reciprocals of Natural Numbers? [duplicate]

I want to calculate the sum of first $n$ natural numbers. I used the following C program to compute the first '$n$' digits : ...
2
votes
3answers
56 views

Euler sum of a divergent series

So I have a series $1+0+(-1)+0+(-1)+0+1+0+1+0+(-1)+...$ Is it correct to rearrange this as $1+0+(-1)+0+1+0+(-1)+0+1+0+(-1)+0...$ The second problem can be done as an Euler sum and the answer is ...
3
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1answer
24 views

Convergent series? Gamma/power function

Is it true to use as a general rule of thumb that the Gamma function always "kills" power function in a series? I mean: $$\sum_{n=1}^{\infty} \frac{C^n}{\Gamma(n)^p}<\infty$$ no matter the constant ...
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1answer
46 views

Some clarifications on analytic continuation of Riemann's Zeta function on $\frac 1 2$

Here's my problem: Riemann's Zeta function converge iff $x>1$ so if I want to have a finite value for $\zeta(\frac 1 2)$ I need to use it's analytic continuation but Riemann's hypothesis states ...
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0answers
44 views

In the real world of nature, is the sum of all the positive integers really -1/12? [duplicate]

Is it true that the infinite series 1+2+3+4+...= -1/12? In the equations of a physics theory, if you replace that series with the number -1/12, will your results match real world experimental data? ...
0
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1answer
115 views

Is my $1+1+1+1+1…=-\frac{1}{2}$ proof correct?

Let $x = 1+1+1+1+1+1 ...$ Let $y=1-1+1-1+1-1 . . .$ First, let's find the value of $y$. The partial sums of $y$ are $s_n=(1,0,1,0,1,0,...)$ If you take the means of the partial sums, you will get ...
7
votes
4answers
132 views

Does this series $\sum_{i=0}^n \frac{4}{3^n}$ diverge or converge?

I a newbie to series, and I have not done too much yet. I have an exercise where I have basically to say if some series are convergent or divergent. If convergent, determine (and prove) the sum of the ...
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1answer
32 views

Cesàro means of divergent series

Does $\sum \limits_{n=2}^\infty n$ have a greater Cesàro mean than $\sum \limits_{n=1}^\infty n$? If not, then is there any other sense of "mean" in which the former's mean is greater than the ...
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2answers
47 views

determine the values that series converges

Determine for what values of $x \in \Bbb R$ the series $$\sum_{n = 1}^\infty \frac{(-1)^n}{2n+1}\left(\frac{1-x}{1+x}\right)^n$$ coverges. I have tried the alternating series test but I don't think ...
0
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2answers
30 views

Convergence/divergence of geometric series when $k = 1$?

Usually, when we try to determine convergence/divergence, we simply find the quotient $k$, and if $|k|<1$ we say the series is convergent. At least this is how the textbooks present it. In set ...
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1answer
23 views

Solution check to Absolute convergence test

By the absolute convergence test, the series $$\sum \frac{(-1)^{n}}{\sqrt{n+1}+\sqrt{n}} =\sum a_{n}$$ diverges since $$\sum \frac{(-1)^{n}}{\sqrt{n+1}+\sqrt{n}} < \sum \frac{1}{n^{\frac{1}{2}}}$$ ...
3
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2answers
61 views

Absolute convergence of $\sin(n)/(n^2)$

Prove that $$\sum_{n=1}^{\infty} \frac{\sin(n)}{{n}^{2}}$$ is either absolutely convergent, conditionally convergent or divergent. Note that $$\sin(n) \in [-1,1] \text { for} \left| ...
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3answers
45 views

Convergence of $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^{1/n}}$

Does the series $\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^{1/n}}$ converge? converge in absolute value or conditionally? It's easy to see that in absolute value the general term tends ...
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3answers
51 views

Does $\sum_{n=2}^ \infty \frac 1 {n \sqrt {ln \ n}}$ converge?

I want to figure out if this sum converges or diverges: $$\sum_{n=2}^ \infty \frac 1 {n \sqrt {ln \ n}}$$ I tried comparing it to the harmonic series, but this is less than that so it was no use. The ...
0
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3answers
117 views

Give an example of a divergent series $\sum a_n$ for which $\sum a^2_n$ also converges.

So, I know an answer to this problem is $a_n = \frac{1}{n}$, but can someone explain to me WHY that's the answer? Also, are there any alternative answers to this problem?
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1answer
35 views

Series convergence/divergence

I'm trying to figure out whether the following series diverges or converges by using D'Alemberts (quotientcriteria), Cauchy (integral- and rootcriteria) and Leibniz convergence test for alternating ...
0
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0answers
25 views

The series $n^{-1-1/n}$ diverges, but how do I show this? [duplicate]

I have problems with the following sum $n^{-1-1/n}$. I need to show that the sum diverges, I know that I have to use the comparison test after spending too much time with the wrong tests. My problem ...
0
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1answer
25 views

show if a sum is uniform convergent

my teacher recommended to use this if I need hints on homework help. I am trying to determine that the series $\sum_{n=1}^\infty \frac{\xi}{n}$ on $\xi \in (0,1)$ converges uniformly or doesn't ...
3
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3answers
64 views

Let $\{ a_n\} $ be a sequence of non-negative real numbers such that the series $ \sum_{n=1}^{\infty}a_n $ is convergent.

Let $\{ a_n\} $ be a sequence of non-negative real numbers such that the series $ \sum_{n=1}^{\infty}a_n $ is convergent. If $p $ is a real number such that the series $ \sum{\frac{\sqrt a_n}{n^p}} $ ...
7
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2answers
150 views

Does $\sum a_n$ converge if $a_n = \sin( \sin (…( \sin(x))…)$

Does $\sum a_n$ converge if $a_n = \sin( \sin (\cdots( \sin(x))\cdots)$, $\sin$ being applied n times and $x \in (0, \pi/2)$? What about $\sum a_n^r$ for $r \in \mathbb{R^+}$?
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1answer
28 views

Divergence of this series

I don't know how to solve this: Fix a positive real number $k$ and set $p=1/k$. For $n \ge 0$, consider the function $f_n : \mathbb{C} \rightarrow \mathbb{C}$ defined by $f_n(z)=k^n z^n$. Fix $z_0 ...
1
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1answer
59 views

Showing divergence for $\sum^\infty_{n=1} \bigl(1-\cos (1/\sqrt n)\bigr)$

Show the divergence of $\displaystyle\sum^\infty_{n=1} \left(1-\cos\frac 1 {\sqrt n}\right)$ My attempt: Since $\sin x\in [-1,1]$ then $\sin\frac 1 {\sqrt n}-\cos\frac 1 {\sqrt n}\le ...
5
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3answers
65 views

About the divergence of $\sum^\infty_{n=1}\frac 1 {n\cdot n^{1/n}}$

There's a rule of thumb that if $a>1$ then the series: $\displaystyle\sum^\infty_{n=1}\frac 1 {n^a}$ converges. Now the series: $\displaystyle\sum^\infty_{n=1}\frac 1 {n\cdot n^{\frac 1 ...
5
votes
4answers
78 views

Convergence of $\sum^\infty_{n=1}\arctan(\frac 1 {\sqrt n}) $ and how to approach trigonometric expressions in sums

Does $$\sum^\infty_{n=1}\arctan\left(\frac 1 {\sqrt n}\right)$$ converge? The series probably diverges and I should probably use the comparison test, but I don't know what to use. Note: no integral ...
3
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2answers
82 views

Why doesn't $\sum_{n=1}^\infty \frac{1}{n^{1+\frac{1}{n}}}$ converge?

$\sum_{n=1}^\infty \frac{1}{n^{1+\frac{1}{n}}} = \infty$. Is there a comparison that works well to prove this?
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2answers
44 views

Using the (limit) comparison test to test $\sum\limits_{n=1}^\infty\sin\frac1n$ for con-/divergence

Problem: Use the comparison test, or limit comparison test, to see if $$\sum\limits_{n=1}^\infty\sin\frac1n$$ converges or diverges. My attempt: Sadly empty. So far, I've only dealt with sums where ...
0
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1answer
54 views

Series and Absolute Convergence

I was wondering if I could get a hint, and a hidden answer on these two series. We are suppose to find out if they converge absolutely, or conditionally. I am stuck on the test I should use. (1) ...
23
votes
3answers
628 views

The series $\sum_{n=1}^\infty\frac1n$ diverges

We all know that the following harmonic series $$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots $$ diverges and grows very slowly!! I have seen many proofs of the result but ...
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votes
4answers
50 views

Is $\frac{1}{\sqrt x}$ a converging or diverging sequence?

Suppose that $\{\epsilon\}$ is a sequence of positive reals converging to 0. Check whether the set $\{n\epsilon \}$ is bounded. I am looking for the answer of this question.
2
votes
2answers
54 views

Baby Rudin Chapter 3 Problem 11(d)

Suppose that $a_n > 0$ for all $n \in \mathbb{N}$ and that $\sum_{n=1}^\infty a_n = +\infty$. Let $b_n \colon= {a_n \over {1+na_n}}$ for all $n \in \mathbb{N}$. Then we can show the following ...
4
votes
1answer
40 views

Topology for Divergent Sequence

Let $a_n$ be a sequence of real numbers and define the "Cesàro limit" of $a_n$ to be $C \lim\limits_{n\rightarrow \infty}{a_n} = \lim\limits_{N\rightarrow\infty}{\frac{1}{N}} ...