Questions on whether certain series diverge, and how to deal with divergent series using summation methods such as Ramanujan summation and others.
2
votes
1answer
123 views
Summability of divergent series with binomial coefficients
I have been looking in the known literature before to ask this question that could have a very easy answer. Let me state the problem. I have a series like this
$$(1-x)^\alpha=
...
7
votes
2answers
255 views
Divergent Series
Thinking about divergent series and ways of "summing" them, they seem to fall into two categories (roughly):
Series like $\sum_{k=1}^\infty \frac{1}{k}$, which defy all kinds of regularization or ...
12
votes
3answers
566 views
If $\sum_{1}^{\infty}(a_n)^3$ diverges, does $\sum_{1}^{\infty}(a_n)$?
Per the title, if $\sum_{1}^{\infty}(a_n)^3$ diverges, does this imply that $\sum_{1}^{\infty}(a_n)$ diverges?
I'd appreciate hints (!) for dealing with this excercise.
EDIT Per the contrapositive, ...
11
votes
1answer
224 views
Regularizing divergent series and Bernoulli numbers
Here is the actual problem I need a proof for: (I made an error writing down the equation initially)
$$B_{k+1} = \frac{(-1)^k}{2^{k+2}-2} \sum_{q=0}^{k} \binom{k+1}{q} 2^q B_q$$
Below is my ...
2
votes
0answers
159 views
Is $\sum \limits_{n=0}^{\infty}2^n=-1$? [duplicate]
Possible Duplicate:
Infinity = -1 paradox
MinutePhysics has what initially looks like a divergent series summing to -1. The youtube comments are... lacking in clarity. The argument ...
4
votes
1answer
67 views
Prove that $\sum_{n=1}^{\infty}x^n\frac{(n!)^3}{(3n)!}$ converges when $|x|$ < 27 and diverges when $|x| > 27$
This is a homework question that I am stuck on... I am not sure which test to use to prove this statement. If someone could let me know at least which test to use to push me in the right direction ...
5
votes
2answers
130 views
Changing divergent series to convergent by re-ordering denominators
Suppose $a_n$ is strictly decreasing and positive and $\sum_{n>1}a_n/n=\infty$, let $g:\mathbb N\to\mathbb N$ be a bijection between the positive integers, can we have ...
12
votes
1answer
322 views
Is it possible to assign a value to the sum of primes?
It is possible, by means of zeta function regularization and the Ramanujan summation method, to assign a finite value to the sum of the natural numbers (here $n \to \infty $) :
$$ 1 + 2 + 3 + 4 + ...
5
votes
1answer
250 views
For bounded sequences, does convergence of the Abel means imply that for the Cesàro means?
See the title. This is true if the sequence is nonnegative; some Tauberian theorems which I was able to find give some more general sufficient conditions. I would like to know if this is true for ...
7
votes
2answers
178 views
Does the family of series have a limit?
For $r<1$ define $F(r)=\sum_{n\in\mathbb N}(-1)^nr^{2^n}$. Does $F$ have a limit as $r\nearrow 1$?
2
votes
1answer
146 views
On the limit of a product of two sequences
If $b_n$ is decreasing and positive and $\sum b_n$ converges and $1/a_n$ is decreasing and positive and $\sum 1/a_n$ diverges, must $\lim \limits_{n \to \infty} a_n b_n=0$?
3
votes
1answer
155 views
Dense subseries of divergent series
Suppose $\sum_{n>1} a_n=\infty$ and $0<a_{n+1}<a_n$.
Let $b_k=a_k$ or $b_k=0$ for all integers $k$.
Let $R=\lim_{n\rightarrow\infty}((1/n)\sum_{q=1}^{q=n} b_q/a_q)$
If $R>0$, how to show ...
0
votes
2answers
86 views
Trying to revert a series with problematic log term
I'm stuck on a problem which I'm not sure has a solution. I have the first few terms of a series I want to invert,
$y(x)=\ln(x)+a_0+\frac{a_1}{x}+\frac{a_2}{x^2}+\cdots$
I know the inverse exists ...
1
vote
1answer
107 views
Divergence of subsequences
How can we show that if $f:\mathbb{N} \rightarrow \mathbb{N}$ with $f(n+1) \geq f(n)>1$ for all $n\ge 1$ and $\sum_{n = 1}^\infty \frac{1}{f(n)} = \infty$, then for all integers $k>1$ we have ...
2
votes
3answers
127 views
Slowing down the divergence
Is there any unbounded function $f:\mathbb{N} \rightarrow \mathbb{N}$ with $f(n+1) \geq f(n)>1$, such that for every $g:\mathbb{N} \rightarrow \mathbb{N}$ with $g(n+1) \geq g(n)>1$ and ...
0
votes
1answer
442 views
A question about the interval of convergence for alternating series
Say we are given the simple power series $$\sum_{i=0}^{\infty}(-1)^k\frac{(x-4)^k}{2^k}$$
The interval of convergence can easily be shown to be $x\in(2,6)$ using the Root Test, and, since absolute ...
6
votes
3answers
143 views
For what $t$ does $\lim\limits_{n \to \infty} \frac{1}{n^t} \sum\limits_{k=1}^n \text{prime}(k)$ converge?
The average of all primes is $$\lim\limits_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} \text{prime}(k) ,$$ which diverges.
What is the smallest $r$ such that for $t>r$, $$\lim_{n \to \infty} ...
1
vote
2answers
153 views
Using Leibniz to prove convergence of $\sum\limits_{n=1}^{\infty}(-1)^{n+1}\frac{3^{n}}{3 \cdot 5 \cdot 7 \cdots (2n+1)}$
$$\sum\limits_{n=1}^{\infty}(-1)^{n+1}\frac{3^{n}}{3 \cdot 5 \cdot 7 \cdots (2n+1)}.$$
I am trying to use Leibniz in order to prove that the series converges.
I don't know if I am doing it correctly. ...
1
vote
3answers
262 views
Use the definition of a limit/triangle inequality to show divergence
I just asked a question about this kind of stuff so I feel bad asking again, but I could use some help. This is a homework question that reads:
Use the definition of limit to prove that the sequence ...
13
votes
2answers
464 views
A version of Riemann's theorem
Suppose that series $\sum^\infty_{n=1} u_n=s$ converges conditionally. Then for each $s'\gt s$ there exists a permutation of positive integers $\sigma:\mathbb{N}\to\mathbb{N}$ such that
if ...
5
votes
2answers
236 views
Series Divergence - Apostol Calculus Vol I, Section 10.20 #7
Prove that $\displaystyle\sum_{n=2}^\infty\frac {(-1)^n}{\sqrt{n}+(-1)^n} $ diverges.
It is easy to see that this absolutely diverges, however how can it be proven to diverge in general? The idea as ...
5
votes
4answers
277 views
Proof that series diverges
Prove that $\displaystyle\sum_{n=1}^\infty\frac{1}{n(1+1/2+\cdots+1/n)}$ diverges.
I think the only way to prove this is to find another series to compare using the comparison or limit tests. So far, ...
4
votes
1answer
346 views
A divergent series from Futurama
I was watching Futurama and in a recent episode, the professor creates a duplication machine.
The machine basically took something and then made 2 copies at 60% the size.
Somehow Bender got caught ...
1
vote
4answers
108 views
Diverging sequence
I can't understand diverging sequences. How can I prove that $a_n=1/n^2-\sqrt{n}$ is divering? Where to start? What picture should I have in my mind? I tried to use $\exists z \forall n^* \exists n\ge ...
6
votes
5answers
1k views
Sum of infinite divergent series
I have learned that positive infinity plus negative infinity isn't equal to zero, it's an indeterminate form. However what happens if we subtract two infinite divergent series ...
1
vote
1answer
155 views
Is it possible that these series's value is $0$?
$$\sum_{n=1}^{\infty}\frac{\left ( -1 \right )^n}{n^x}\cos{\left ( y\ln{n} \right )}$$
$$\sum_{n=1}^{\infty}\frac{\left ( -1 \right )^n}{n^x}\sin{\left ( y\ln{n} \right )}$$
$x$ and $y$ are arbitrary ...
2
votes
3answers
668 views
How do I prove the divergence of this series?
How do I prove that $\displaystyle\sum_{n\geq 1}\frac {1}{\ln^2n}$ is a divergent series?
4
votes
2answers
158 views
Convergence of $\sum \frac{a_n}{S_n ^{1 + \epsilon}}$ where $S_n = \sum_{i = 1} ^ n a_n$
Let $a_n$ be a sequence of positive reals, such that the partial sums $S_n = \sum_{i = 1} ^ n a_i$ diverge to $\infty$. For given $\epsilon > 0$ do we have $$\sum_{n = 1} ^ \infty \frac{a_n}{S_n^{1 ...
9
votes
3answers
230 views
Many convergent sequences imply the initial sequence zero?
In connection to this question, I found a similar problem in another Miklos Schweitzer contest:
Problem 8./2007 For $A=\{a_i\}_{i=0}^\infty$ a sequence of real numbers, denote by ...
4
votes
1answer
109 views
Reverse Erdős-Turan
If $\{a_i\}$ is a set of positive integers which contains arbitrary long arithmetic progressions, how to show that $\sum\limits_{i=1}^{\infty}\frac{1}{a_i}=\infty$?
5
votes
1answer
303 views
Are there complex Bernoulli numbers?
I am aware of the generalized Bernoulli numbers, but these are not what I'm looking for. I was wondering if there exists such a thing as fractional, real or even complex Bernoulli numbers ( $B_z$ for ...
39
votes
4answers
2k views
Double sum - Miklos Schweitzer 2010
There is a question in the Miklos Schweitzer contest last year that keeps bugging me. Here it is:
Is there any sequence $(a_n)$ of nonnegative numbers for which $\displaystyle\sum_{n \geq 1}a_n^2 ...
5
votes
1answer
325 views
Exercising divergent summations: $\lim 1-2+4-6+9-12+16-20+\ldots-\ldots$
I'm trying to make sense of some (assumed to be) simple exercises in divergent summation. One example I cannot resolve.
First I assume the sequence of binomialcoefficients $ \{ b_k = \binom k2 ...
34
votes
5answers
3k views
Why does $1+2+3+\dots = {-1\over 12}$?
$$\lim_{N\to\infty} \sum_{i=1}^N \, i = +\infty$$
$\displaystyle\sum_{n=1}^\infty n^{-s}$ only converges to $\zeta(s)$ if $\text{Re}(s) > 1$.
Why should analytically continuing to $\zeta(-1)$ ...
12
votes
2answers
1k views
Series of logarithms $\sum\limits_{k=1}^\infty \ln(k)$ (Ramanujan summation?)
I had this question earlier, so to say as a "standalone" problem, but now it pops up in context of an analysis with the lngamma-function. As well as we can convert the question of sums of like powers ...
2
votes
1answer
228 views
Geometric Series and its infinite sum
when we say $1+r+r^2+r^3+\cdots=S$ do we mean $S$ is the partial sum of first $n$ terms of this geometric series where $n$ goes to infinity (which is not boundless due to the existence of $n$th term) ...
8
votes
2answers
390 views
Values of the Riemann Zeta function and the Ramanujan Summation - How strong is the connection?
The Ramanujan Summation of some infinite sums is consistent with a couple of sets of values of the Riemann zeta function. We have, for instance, $$\zeta(-2n)=\sum_{n=1}^{\infty} n^{2k} = 0 ...
6
votes
1answer
189 views
Nitpicky question about harmonic series
I wanted to prove as much as I could about the rate of divergence of the harmonic series without resorting to textbooks; I did this by checking a little computationally and using that as motivation ...
1
vote
3answers
216 views
Why do we pick $\frac{1}{1-x}$ for $f(x)=1+x+x^2+x^3+…+x^n$, where $n$ tends to infinity?
When we consider the function $f(x)=1+x+x^2+x^3+...+x^n$ where $n$ tends to infinity, we can rewrite this as $$f(x)=1+x(1+x+x^2+x^3+...)=1+x(f(x))\qquad (1)$$ After some algebraic manipulations, we ...
