Questions on whether certain series diverge, and how to deal with divergent series using summation methods such as Ramanujan summation and others.

learn more… | top users | synonyms

-1
votes
0answers
24 views

Using Cauchy's convergence test on a specific series

This is my series: Series WolframAlpha tells me that this series converges. Here's what I get instead with the Cauchy convergence test: My solution What am I doing wrong exactly?
1
vote
1answer
18 views

Divergence of squared sum of Chebyshev Polynomials $\equiv L+R$ has empty point spectrum

The Chebyshev Polynomials of the second kind $U_n$ are the solutions of the differential equation $$(1-x^2)U_n''(x)-3xU_n'(x)+n(n+2)U_n(x)=0$$ Alternatively they are defined inductively: $$U_0(x)=1 ...
2
votes
1answer
31 views

Finding the maximum value of a divergent series [on hold]

I came across this divergent sum- $$\sum_{n=1}^\infty\frac{1}{n+1}$$ Now,a divergent sum does not a limit.So is it possible to get a maximum value for the sum or more specifically prove that ...
2
votes
0answers
42 views

In what sense 1 + 0 + 2 + 0 + 3 + 0 + … = 1/24?

We know that the series $$ 1+2+3+\cdots=-\frac{1}{12} ~~ (1) $$ and $$ 0+1+0+2+0+3+\cdots=-\frac{1}{12} ~~ (2) $$ belong to the elementary Ramanujan class $R=4$ (definition, also here) and the series ...
1
vote
4answers
73 views

What is the difference between sum and integral?

I am a beginner in calculus and I want to know what is the difference between sum and integral. More specifically I came across this example: Compare $$\sum^\infty_1\frac1x\space \text{and} \space ...
-1
votes
2answers
114 views

What $+1+1+\cdots$ really equals

$1+1+1+\cdots$ is clearly a divergent series, so you'd say that it tends towards infinity? Through analytic continuation of the zetafunction the value $-1/2$ could be assigned the sum, right? But if ...
-1
votes
0answers
60 views

Proof of 1^2 + 2^2 + 3^2 + 4+2 +… infinity = 0 [closed]

Leonhard Euler proved that the sum of the series $$ 1^n + 2^n + 3^n + 4^n + ... ∞ = 0 $$ where n is an even number. I know that this is a diverging series, but the infinity part of it somehow makes ...
0
votes
0answers
31 views

Show that 1 + 2 + 0 + 4 + 0 + 0 + 0 + 8 + … = -1.

The diluted series of powers of $2$ $1+2+0+4+0+0+0+8+\cdots$ belongs to the elementary Ramanujan class $R=2$ and is summable to $-1$ (definition, also here). How to prove that result given ...
2
votes
2answers
59 views

Why is $\sum_{n=1}^\infty {1\over{n^{1+ {1\over \ln n}}}}$ divergent?

I am just starting to learn Calculus. If anyone could help me that would be very useful. Thanks ahead From here: how to prove $\sum {\frac{1}{n^{1+1/n}}}$ is divergent I don't really get how to use ...
1
vote
0answers
63 views

Assigning values to a divergent integral?

Question If I can assign the series of the zeta function to: $$ \zeta(-1) \to 1+2+3+\dots$$ why can't we assign the integral $$ \int_{0}^{\infty} x dx \to 0$$ and it still have some physical ...
1
vote
1answer
31 views

Series generated by food donation

Not counting the fact that students need to take food from the shelves: Assume that I donate $1 $ pound of food and I get $3 $ of my students to donate, but they only donate $35\% $ of what I gave. ...
1
vote
1answer
44 views

What use does a Cauchy principal value and divergent summation have?

Through some questionable methods, there lies an answer to the following integral. $$\int_{-a}^a\frac{dx}x=0$$ You may question its soundness at first glance since: ...
2
votes
4answers
265 views

Can we assign a value to the sum of the reciprocals of the natural numbers?

I know the sum of the reciprocals of the natural numbers diverges to infinity, but I want to know what value can be assigned to it. ...
0
votes
2answers
32 views

Problems using D'alemberts Ratio test for convergence or divergence

The geometric series is as follows : $$n/2^n$$ I am using the ratio test therefore comparing : $$(n+1/2^{n+1})/ (n/2^n)$$ my next line of work is : $$(n+1/2^{n+1}) * (2^n/n)$$ however I am not ...
0
votes
3answers
32 views

Prove wether or not the following series diverges or converges: $\sum_{n=0}^\infty {(-1)^nn\over n+1}$

Prove wether or not the following series diverges or converges: $\sum_{n=0}^\infty {(-1)^nn\over n+1}$ I am just not sure, I know if I use the absolute value test for convergence and root test it ...
3
votes
0answers
42 views

Can all series in the elementary Ramanujan class R = 2 be shifted?

For $f(x)=\sum_{n=0}^\infty a_nx^n$ and $g(x)=f(x)-Rf(x^2)$, $R\neq1$, $f(1)$ belongs to the elementary Ramanujan class $R$ if $g(1)$ is Abel summable. The elementary Ramanujan sum of $f(1)$ is ...
-2
votes
1answer
33 views

Convergence of infinite sum including cosh functions?

I am attempting to code up an equation that includes an infinite sum of cosine and hyperbolic cosine functions, namely: $$ \sum_{m=0}^{\infty} \frac{ \cos[(2m+1)\pi x/s] \cosh[(2m+1)\pi z/s] } ...
2
votes
3answers
182 views

Why $\zeta (1/2)=-1.4603545088…$?

I saw $\zeta (1/2)=-1.4603545088...$ in this link. But how can that be? Isn't $\zeta (1/2)$ divergent since ...
3
votes
1answer
95 views

Is $\lim_{n\to\infty}\frac{n(a_n-a_{n+1})}{a_{n+1}}=0$ if $\sum_{n\geq 1} a_n/n$ diverges?

Consider a sequence $(a_ n)$ of positive and monotonically decreasing terms for which $\sum_{n\geq 1} a_n/n$ diverges and $a_ n\to 0$. I claim that that ...
1
vote
1answer
124 views

Why a + b + c + a + b + c + a + b + c + … = (a - b - 3c)/6?

Given that $1+1+1+\cdots=-1/2$, why the sum of $a+b+c+a+b+c+a+b+c+\cdots$ is $(a-b-3c)/6$ instead of $-(a+b+c)/2$? Edit. What about a general periodic series? $$ a_1+\cdots +a_N+a_1+\cdots ...
8
votes
3answers
163 views

Does this series diverge if $a_n\to 0$?

Consider a given sequence $\{a_n\}$ of positive terms and the series $$\frac {a_ 1^2}{a_1}+\frac{a_2^2}{a_1+a_2}+\frac {a_ 3^2}{a_1+a _2+a_3}+\cdots.$$ I can show that this series diverges if the ...
1
vote
1answer
33 views

Division between power series that converge at least for |x| < r, only valid for |x| sufficiently small?

I have this book Calculus, Ninth Edition by Varberg, Purcell, and Rigdon; there's a particular point of a theorem (and another line after that) about Infinite Series that I really don't understand. I ...
21
votes
1answer
553 views

Is this a way to prove there are infinitely many primes?

Someone gave me the following fun proof of the fact there are infinitely many primes. I wonder if this is valid, if it should be formalized more or if there is a falsehood in this proof that has to do ...
0
votes
2answers
43 views

Is convergent or divergent $\sum_{n=1}^\infty{(-1)^n\dfrac{\ln{n}}{n\ln{\ln{n}}}}$?

$$\sum_{n=1}^\infty{(-1)^n\dfrac{\ln{n}}{n\ln{\ln{n}}}}$$ Any suggestions? I tried absolute convergence, but it doesn't work.
-2
votes
4answers
78 views

Is this series convergent or divergent $\sum_{n=1}^\infty{\frac{(2n)!}{n^{2n}}}$? [closed]

I have problem with $$\sum_{n=1}^\infty{\dfrac{(2n)!}{n^{2n}}} $$ I try to use Cauchy Condensation Test, but unsuccessfully. Any suggestions? Thanks for any help.
1
vote
3answers
53 views

State whether the following series converges or diverges $\sum\limits_{n=0}^\infty{7^n - 2^n\over(2n)!}$ [closed]

$\sum\limits_{n=0}^\infty{7^n - 2^n\over(2n)!}$ Thanks in advance for any help you are able to provide.
5
votes
1answer
88 views

Functions on real line which preserves dfferent modes of convergence and preserves divergence of real infinite series

From this question The set of functions which map convergent series to convergent series , it is known that the set of functions on real line which maps convergent series to convergent series is ...
2
votes
3answers
39 views

Divergence of $\sum_{n\geq 2} \frac{1}{\ln^p n}$ for $1<p\leq \infty$ [closed]

Can anyone help me to prove that $(x_n)\notin l_p$ with $x_n=\frac{1}{\ln^p n}$? Suppose $1< p<\infty$.
3
votes
0answers
79 views

Is 1-1+2-2+3-3+… Ramanujan summable?

The elementary Ramanujan sum of the series 1-1+2-2+3-3+... is 1/8 (definition). Is this series Ramanujan summable according to Hardy's definition? Edit. For $$ f(x)=x-x^2+2x^3-2x^4+3x^5-3x^6+\cdots, ...
1
vote
2answers
56 views

Does $\sum_i \frac{b_i^2}{i}$ diverge assuming the Cesaro mean of $\{b_i\}$ is greater than zero?

Consider a sequence of positive numbers $\{b_i\}$ bounded above. Suppose that the Cesàro mean of these numbers is positive. That is, $$\lim_{n\to\infty}\frac{b_1+\dots+b_n}{n}=b>0.$$ Can it be ...
0
votes
0answers
11 views

Euler summation identity $E_{y_1}E_{y_2}=E_{\frac {y_1y_2}{1+y_1+y_2}}$

I am not exactly sure how many types of Euler summation there are. I read one paper that explained how it is related to the Bernoulli polynomials, and now, on Wikipedia, I found this separate article: ...
2
votes
3answers
41 views

Real-analysis: convergence of sequence and convergence of series

True or false (if true, prove it otherwise give an counterexample). "Suppose $(a_n)$ is a sequence such that $\lim_{n\to\infty}a_n=\alpha$ with $\alpha \in (0,1)$. Then the series ...
0
votes
0answers
16 views

Uniform and ordinary convergence of a series.

I have a series of the kind $\sum\limits_{n=1}^{\infty} \frac{a_nt^n}{1+t^{2n}}$, $t\in(0,\infty)$. Btw, $a_n$ are real and they are determined by several integrals, but it is possible to compute any ...
-4
votes
3answers
81 views

Series divergent or convergent?

Given that $a_n\rightarrow\frac{1}{3}$ as $n\rightarrow\infty$, does the series $a_{k+1}-a_k$ converge or diverge?
4
votes
2answers
39 views

Every convergent sequence is a Cauchy sequence.

Today, my teacher proved to our class that every convergent sequence is a Cauchy sequence and said that the opposite is not true, i.e. Not every Cauchy sequence is a convergent sequence. However he ...
1
vote
3answers
59 views

Does the series $\sum_{n=2}^\infty {\frac{n+2}{n^3-2n^2+1}}$ converge?

Do the following converge: $\sum_{n=2}^\infty {\frac{n+2}{n^3-2n^2+1}}$ For this one I think the answer is no I just can't prove it. I split it up into partial fractions and got: ...
1
vote
1answer
33 views

Is this property of two sequences $a_n$ and $b_n$ possible?

Is it possible that two sequences ${a_n \over b_n} \to 1$ but $| a_n - b_n | \to \infty$? This question occured to me while reading the Wikipedia page on the prime counting function, $\pi(n)$. It's ...
0
votes
2answers
69 views

Does the series $\frac{\cos(\pi n)}{n}$ converge?

Does the series $\left\{{\cos(\pi\cdot n)\over n}\right\}$ converge? I think it does, but can't find a series convergence test that applies to it. I can't compare it to the series $\left\{{1\over ...
0
votes
1answer
44 views

Convergence/divergence of series with sinus

I want to know if this converges or diverges. $\sin(n*\frac{\pi}{2})\frac{n^2 + 2}{n^3+n}$ I solved that $\frac{n^2 + 2}{n^3+n}$ diverges by limit comparasion test with $\frac{1}{n^2}$ but I dont ...
1
vote
4answers
54 views

Analysis: Proving divergence using partial sums

How do I prove that $$\sum_{n=1}^{\infty}\frac{1}{n^s}$$ diverges for $s<1$, by estimating its partial sums?
2
votes
2answers
50 views

Sequence Converge or Diverge

Does the following sequence converge or diverge: $a_n=\frac{\sin{\left(n\right)}}{2^{n}}$? My initial thought was that any value of $n$ to $\sin$ will be less than $1$.
2
votes
3answers
32 views

Pointwise convergence of $\frac{x^n}{1+x^n} $

Is this sequence convergent or divergent? I first thought it was convergent due to pointwise limits existing at every value of $x$ but now im not sure wether or not I am right. My sequence is ...
1
vote
3answers
44 views

Proving convergence of $\sum_{n=1}^\infty n^{\ln(x)} $

I want to prove that $\sum_{n=1}^\infty n^{\ln(x)} $ is convergent for x in $(0, \frac {1}{e})$ interval and divergent for $ x \geq \frac {1}{e} $. I am lost on how to prove it. Could someone please ...
0
votes
1answer
34 views

Testing series for divergence

I have the following series: $$\sum_{n=1}^\infty \ln(2(n+1))- \ln(2n)$$ How can I test it to show it is divergent?
0
votes
1answer
24 views

method in finding absolute convergence of a given series

How do I show absolute convergence for the series $$\sum_{n=0}^{\infty} \frac{n}{\sqrt{2n^5 +1}}$$ I have already showed by Comparison test that it is convergent. I am after the way of showing $\sum ...
5
votes
1answer
83 views

Prove That $\displaystyle\sum_{n=1}^\infty \sin(n^p)$Diverges For All $p>0$

Prove that the series $\displaystyle\sum_{n=1}^\infty \sin(n^p)$diverges for all $p>0$. This should be simple but I have been failing... My latest attempt is Cauchy's criterion.
1
vote
1answer
19 views

Convergence according to $n$th-term test

According to the $n$th-term test for divergence, a series $\sum_{n=1}^\infty a_n$ diverges if $\lim_{n\to\infty} a_n \neq 0$. But I don't actually get it. I thought that if I have $$ ...
1
vote
2answers
84 views

Convergence of $\sum_{n = 1}^\infty 1/n^2$.

I know that $\sum_{n=1}^\infty 1/n$ diverges whereas $\sum_{n=1}^\infty 1/n^2$ converges. Intuitively, I do not see the difference. If $n \to \infty$, the denominators in both fractions will be so ...
1
vote
2answers
29 views

Convergence of a sum using Leibniz test

I tried to use ratio and root test to see the convergence of $$\sum_{n=1}^{+\infty}\frac{(-1)^{n}3n}{4n-1}$$ but both were inconclusive. I also tried to use Leibniz test. I got that ...
1
vote
3answers
53 views

Proof of $\sum \frac 1 {p_{2n}} = \infty$

I found the following Theorem Let $p_n$ denote the $n$-th prime number. $S_1= \sum_{n \in \Bbb N} \frac 1 {p_{2n}} = \infty$ and $S_2=\sum_{n \in \Bbb N} \frac 1 {p_{2n+1}} =\infty$. Proof ...