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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4
votes
2answers
322 views

Does this sum converge, is my solution good?

$$ \sum_{n=1}^\infty \frac{\sin(n)^{7}}{(n^{7}+1)^{1/2}} $$ I would say that it doesn't converge, cause I would write this as: $$ $$ $$ \frac{\sin(n)^{7}}{(n^{7})^{1/2}} $$ when $$ \lim_{n\to ...
1
vote
2answers
38 views

Convergence of series 4

Determine if the following series is convergent or not: $$\frac{1}{\sqrt{n} \log n}$$ I tried: $a_k = \frac{1}{\sqrt{n} \log n}$ $b_k= \frac{1}{\sqrt{n}}$ then did: $\frac{a_k}{b_k}$ and got ...
-4
votes
1answer
22 views

Absolutely convergent, conditionally convergent or divergent [on hold]

I have this question: $$\sum_{n=1}^\infty \frac{\cos\left(\frac{n\pi}{12}\right)}{n\sqrt n} $$ How do I figure out if it's absolutely convergent, conditionally convergent or divergent?
1
vote
2answers
54 views

Show that this series is divergent using the comparison test.

Let $\langle a_n\rangle$ be a sequence of positive numbers. Consider $\sum_{n=1}^\infty{Y_n}=\sum_{n=1}^\infty{\left(\frac{a_n}{n}+\frac{n}{a_n^2}+\frac{a_n}{n^3}\right)}$. Show this diverges using ...
-3
votes
0answers
27 views

help with showing a series is divergent [closed]

I tried unsuccessfully to show by convergence tests that the series $$\sum_{n=1}^\infty{\ln^nn\over n^2}$$ is divergent , cant seem to find a way. help would be very appreciated , thanks in ...
3
votes
0answers
41 views

Elemenatry topological proof of Erdos conjecture on Arithmetic Sequences

Define $C_n(A) = \{a \in A : \forall d \in \Bbb{N}$ one of $a + d, a +2d, \dots a + (n-1)d$ is not in $A\}$ For $n \leq m$, we have $C_n(A) \subset C_m(A)$. Proof. Let $x$ be in the LHS. Then ...
1
vote
0answers
33 views

Fibonacci Sequence (conceptual/mechanical)

I'm in Calc II and have my exam on series/sequences tomorrow. There will be an extra credit question related to Fibonacci and am trying to gather as much info on the sequence as possible. -If the ...
0
votes
0answers
18 views

Expression about $\sum_{d=0}^{\infty} e^{dt}\frac{(5d)!}{(d!)^5}5\sum_{n=0}^{\infty} \left( \frac{1}{n+d+1} - \frac{1}{n+5d+1} \right)$ [Done]

The purpose of my question in (here)[converge value of series $\sum_{n=0}^{\infty} \left( \frac{1}{n+d+1} - \frac{1}{n+5d+1} \right) $ was actually to re-expression of following computation. ...
5
votes
1answer
55 views

converge value of series $\sum_{n=0}^{\infty} \left( \frac{1}{n+d+1} - \frac{1}{n+5d+1} \right) $

\begin{align} \sum_{n=0}^{\infty} \left( \frac{1}{n+d+1} - \frac{1}{n+5d+1} \right) = \sum_{n=0}^{\infty}\frac{4d}{(n+d+1)(n+5d+1)}= ? \end{align} I know from the $p$-test, ($i.e$ $\sum \frac{1}{n^p}$ ...
7
votes
2answers
109 views

In what sense does $\sum_{k=0}^{\infty} 2^{2k} = - {1 \over 3}$?

In The Road to Reality Penrose remarks on an identity written down by Euler which is "obviously wrong" and yet correct "on some deeper level". He makes reference to the series again when discussing ...
0
votes
1answer
44 views

Does this summation (involving binomial) have a closed form? If so, what is it?

The following sums are the ones I'm interested in: $\sum_{i=m}^{\Omega}{\binom{i}{m}i^{-k}}$ $\lim_{\Omega\rightarrow\infty}{\sum_{i=m}^{\Omega}{\binom{i}{m}i^{-k}}}$ I already know that ...
2
votes
1answer
47 views

Is the sum of the reciprocals of the squarefree numbers divergent or convergent?

I just come across this question by trying to analyze the pseudoinverse of some infinite matrix (the matrix T as interpreted in my answer to this MSE-question), where this series occurs from some ...
-2
votes
1answer
41 views

Is the following series convergent or divergent. If convergent find the sum if possible. [closed]

How do you find if this series is convergent or divergent? $$\sum_{n=1}^{\infty} \dfrac{\ln(n^2+1)}{\ln(n^3+1)}$$ Is someone able to solve this with all the steps?
-5
votes
1answer
56 views

Partial sum of all natural numbers [duplicate]

I'm pretty sure this question has been asked several times however,... It's obvious that the series $S_n = 1+2+3+4+\cdots +n$ is divergent. However there are many other claims that the sum is -1/12 ...
1
vote
1answer
35 views

$\sum a_{n}$ is convergent, $\sum a_{n}^2$ is divergent. Prove $\sum a_{n}$ is conditionally convergent.

Suppose that the series $\sum_{n=1}^{\infty}a_{n}$ converges while $\sum_{n=1}^{\infty}a_{n}^2$ diverges. Prove that $\sum_{n=1}^{\infty}a_{n}$ converges conditionally.
0
votes
1answer
45 views

Proof of divergence of a series 4

Let $ (a_k)_{k\in\mathbb{N}}$ be a decreasing sequence of positive real numbers. We suppose that there exists a $b>0$ such that $a_k \geq \frac{b}{k}$ for infinite values of $k$ . Prove that the ...
1
vote
2answers
72 views

Proof infinite series $1 + 2 + 3 +\cdots +n+\cdots$ diverges

Question: By considering the partial sums for S, that is $S_n = 1 + 2 + 3 +\cdots +n$ show that the infinite series S does not converge. My answer : I tried to attempt this question, ...
3
votes
5answers
135 views

How do you solve for the limit of this series?

I need to take the limit of this summation so that I kind find out whether it converges or diverges. The equation is: $$\sum_{k=1}^\infty \frac{4}{k+4}$$ What I have tried so far is the following: ...
0
votes
2answers
49 views

Convergence of $\sum_{n=1}^{\infty} \frac{n^3}{\ln(2)^n}$

would the sum from 1 to infinity of $\frac{n^3}{(\ln{2})^n}$ converge? In the limit n tends to infinity the denominator grows more quickly and so the terms go to zero. Using the ratio test I get ...
0
votes
1answer
29 views

Discuss whether the series $\sum \left[(\pi/2)^a - (\arctan n)^a\right]$ converges or not, based on the value of $a$

$$\sum_{n=1}^\infty {\left[ {{{\left( {\frac{\pi }{2}} \right)}^a} - {{(\arctan n)}^a}} \right]} $$ I proved that the series diverges for $a < 0 $ and that the series converges for $a = 1$ (using ...
1
vote
2answers
63 views

Divergent, convergent series

Let $p$, $q \in \mathbb{R}$ and see the series $$ \sum_{n=2}^{\infty} \frac{1}{n^p(\ln n)^q} $$ View with the comparison criterion that if $p> 1$ then the series is convergent for all $q$, and ...
0
votes
0answers
39 views

What to do when Ramanujan summation diverges too?

While using Ramanujan summation to some kind of divergent series I got stuck: let's take the definition of this sum for the terms of a general function $f(x)$: $$\Re(x)=\int_n^xf(t)dt-\frac ...
0
votes
3answers
55 views

Is $\sum_{k=-\infty}^{0}f(-k)=\sum_{k=0}^{\infty} f(k)$ always?

I wonder, whether it is always the case $$\sum_{k=-\infty}^{0}f(-k)=\sum_{k=0}^{\infty} f(k)$$ in regards of summation methods for divergent series?
0
votes
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 ...
-1
votes
4answers
63 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} ...
0
votes
1answer
32 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.
0
votes
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
vote
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
vote
1answer
42 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, ...
0
votes
2answers
23 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
42 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
40 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 ...
1
vote
4answers
42 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? $$d_n=\left(1+\frac{2}{n}\right)^n$$
1
vote
1answer
79 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
votes
2answers
76 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 ...
1
vote
1answer
39 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 ...
0
votes
1answer
21 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
votes
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
58 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
votes
1answer
27 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 ...
0
votes
1answer
51 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 ...
0
votes
1answer
117 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
144 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 ...
1
vote
1answer
37 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 ...
2
votes
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
votes
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 ...
0
votes
1answer
29 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
votes
2answers
66 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| ...
0
votes
3answers
46 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 ...
0
votes
3answers
59 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 ...