Tagged Questions
1
vote
3answers
54 views
Showing that $\sum_{n=3}^\infty\frac1{n(\log n)(\log\log n)}$ diverges
I know that the series diverge, I'm just having hard time showing it.
$$\sum_{n=3}^\infty\frac1{n(\log n)(\log\log n)}$$
Thanks in advance
6
votes
4answers
112 views
Sum of kth roots ($\sum\sqrt[k]{m}$)
I'm trying to find an asymptotic to $$S(n) = \sum_{k=1}^n\sqrt[k]{m}$$ From computational tests, it seems to grow nearly as slowly as $n$. However even $$\sum_{k=1}^\infty\sqrt[k]{m}-1$$ diverges (for ...
7
votes
1answer
48 views
Divergent series of positive numbers
So I've been trying to figure out how to prove the following.
Let $(a_n)$ be a sequence of positive numbers such that $\sum_{n=1}^\infty a_n =\infty$, and define $s_n=\sum_{i=1}^n a_n$. Then ...
1
vote
1answer
51 views
Sequence with convergent subsequences: divergent or convergent?
$(a_n)_{n \in \mathbb{N}}$ be a sequence with the convergent subsequences $(a_{2n}), (a_{2n+1})$ and $(a_{3n})$. Is $(a_n)$ then convergent? Proof or counter-example.
My idea with this question was ...
1
vote
2answers
57 views
ratio test and divergence
I need to be able to prove that this series converges. I know I need to use the ratio test but I do not know how to go about doing it.
Any help is much appreciated! thank you
1
vote
0answers
31 views
Is there an algebra for divergent series summation operators?
Let $D$ denote a divergent series and let $C$ denote a convergent series. Furthermore, let $s : \{ Series \} \to \{ numbers \} $ be a regular, linear divergent series operator, which is either one of ...
0
votes
1answer
21 views
Series Rearrangement
I have two proofs for a real analysis class that I'm struggling with.
The first says,
Prove: $\frac{1}{3n-2}+\frac{1}{3n-1}-\frac{1}{3n}>\frac{1}{3n}$ for $n= 1, 2, ...$ and deduce that ...
3
votes
4answers
66 views
Show that $\sum_{n=0}^\infty n^nx^n$ diverges for $x \neq 0$
Show that the series
$$\sum_{n=0}^\infty n^nx^n$$
diverges for $x \neq 0$.
Any help? I don't know where to start.
2
votes
0answers
61 views
Paradox of Infinity? [duplicate]
If a series such as '$a$' below adds to infinity:
$a = 1 + 2 + 4 + 8 + 16 + \cdots\to \infty$
Multiplying '$a$' by $2$ yields:
$2a = 2 + 4 + 8 + 16 + \cdots\to \infty$
However when I subtract ...
5
votes
1answer
236 views
Missing term in series expansion
I asked a similar question before, but now I can formulate it more concretely. I am trying to perform an expansion of the function $$f(x) = \sum_{n=1}^{\infty} \frac{K_2(nx)}{n^2 x^2},$$ for $x \ll ...
1
vote
2answers
76 views
Does the Intergral Test require the integrand to converge to zero to be applicable?
Is there an error in the following statement of the Integral Test (from David Brannan's Mathematical Analysis)?
The author requires the integrand (equivalently, the general term of the series) to ...
3
votes
2answers
137 views
Convergence of the series $\sum_{n=1}^{\infty}((1/n)-\sin(1/n))$..?
Does the series $\sum_{n=1}^{\infty}((1/n)-\sin(1/n))$ converge..? Can anyone please give me a simple proof..
12
votes
3answers
284 views
Has someone seen a discussion of the (divergent) summation of $\sum\limits_{k=0}^\infty (-1)^k (k!)^2 $?
In extending my studies of the Eulerian matrix and its suitability for a matrix-based divergent summation procedure I'm trying to proceed to sums of the form
$$ S = \sum_{k=0}^\infty (-1)^k (k!)^2 $$
...
2
votes
0answers
76 views
Proof for a summation-procedure using the matrix of Eulerian numbers?
I've discussed a procedure for divergent summation using the matrix of Eulerian numbers occasionally in the last years (initially here, and here in MSE and MO but not in that generality and thus(?) ...
1
vote
0answers
51 views
Given a divergent series, which alterations are convergent?
For a sequence $a_n \gt 0$ I'm given that $\sum a_n$ diverges.
Let $s_n = a_1 + a_2 + ... + a_n$
It's not immediately obvious, but $\sum \frac{a_n}{s_n^2}$ converges, while $\sum \frac{a_n}{s_n}$ ...
2
votes
1answer
39 views
summation of series of powers $ n^{nix} $
is the series ..
$$ \sum _{n=2}^{\infty}n^{kin} $$
here k is a real number
is convergen or divergent ??, for example perhaps we can copare it to the series $$ \sum _{n=2}^{\infty}n^{ix} $$ which is ...
5
votes
4answers
176 views
Construct a convergent series of positive terms with $\displaystyle\limsup_{n\to\infty} {a_{n+1}\over{a_n}}=\infty$
Construct a convergent series of positive terms with $\displaystyle\limsup_{n\to\infty} {a_{n+1}\over{a_n}}=\infty$
My thoughts:
By the theorem: Suppose $a_n\ge0$ for all $n$, and let ...
3
votes
0answers
92 views
For infinite series convergence/divergence: Why doesnt meeting the conditions of the Divergence test imply the Cauchy Convergence Critierion
Assume that the limit of the sequence is zero, $\lim_{n\to\infty}a_n=0$. So its not plainly obvious if the series $\sum a_n$ converges or diverges.
I have wondered for some time. If ...
2
votes
0answers
69 views
On the series $ \displaystyle \sum_{n=1}^{\infty} \frac{\Lambda(n)}{\sqrt{n}} \cos(x \log(n) + a) $.
Is the following series ‘summable’ in the sense that it may be divergent but we can attach a meaning to it?
$$
\sum_{n=1}^{\infty} \frac{\Lambda(n)}{\sqrt{n}} \cos(x \log(n) + a) = (\text{reg}) ...
4
votes
2answers
108 views
Convergent or Divergent
Determine if the series converges or diverges:
$$\sum_{n=1}^\infty \frac{\ln(n)}{n \left|\sin(n)\right|} = $$
We know that
$$
\sum_{n=1}^\infty \frac{1}i = \text{Divergent}
$$
$$
\sum_{n=1}^1 ...
3
votes
1answer
100 views
Can the alternating series test tell me that a series diverges?
My textbook gives this definition of the alternating series test:
Test for alternating series. An alternating series conerges if the absolute value of the terms decreases steadily to zero, that ...
1
vote
2answers
128 views
Convergence or Divergence integral test
$$\sum_{n=1}^\infty\frac1{(\sqrt n + n )\ln (\sqrt n+1)}$$
So im guessing you go $\frac{1}{f(x)}$ and $\frac{1}{g(x)}$
and see if $\frac{1}{f(x)}$ and $\frac{1}{g(x)}$ are both convergent by ...
1
vote
2answers
63 views
if $\sum_{n=1}^{\infty }a_{n}$ converges ($a_{n}$ are non negatives), study the convergence of the serie $\sum_{n=1}^{\infty }\sqrt{\frac{a_{n}}{n}}$
Does it converges? Or it depends on $a_{n}$ terms? I would thank to you if you can explain me if there are some cases in which the serie diverges and other in which the serie converges or if it just ...
3
votes
6answers
237 views
How to prove $ {a_n} = \frac{n!}{2^n}$ diverges to $+ \infty$?
I would like to prove that the sequence $ {a_n} = \frac{n!}{2^n}$ diverges to $+ \infty$. As I understand it, this means that for all numbers $M$, I must find a number $N$ such that for all $n \ge N$, ...
0
votes
4answers
84 views
Another series convergence question
Does this series converge?
$\displaystyle\sum\limits_{n=2}^\infty \frac{\sqrt{n+1}}{(2n^2-3n+1) (\ln n +(\ln n)^2)}$
4
votes
4answers
88 views
Another simple series convergence question
I'm being asked to determine if $\displaystyle\sum\limits_{n=3}^\infty \frac1{n (\ln n)\ln(\ln n)}$ converges. So, using Cauchy's Condensation Test, I reduced the problem to one of determining the ...
3
votes
3answers
101 views
Why does $\sum_{n=1}^\infty (\sqrt[n]{a} - 1)$ diverge for $1 \neq a>0$?
Why does $\sum_{n=1}^\infty (\sqrt[n]{a} - 1)$ diverge for $1 \neq a>0$?
We tried to proof that $\sqrt[n]{a} - 1 > 1/n$, but this doesn't hold. Any ideas?
3
votes
1answer
137 views
Convergence\Divergence of $\sum_{k=2}^{\infty} \frac{\cos(\ln(\ln k))}{\ln k}$
Test for convergence the series
$$\sum_{k=2}^{\infty} \frac{\cos(\ln(\ln k))}{\ln k}$$
My first thought was related to the use of the integral test, but things seem hard.
Could we resort here to some ...
0
votes
1answer
99 views
Determine whether $\sum_{n=1}^{+\infty}c_n$ for a given $c_n$ is divergent or convergent
I want to determine whether the series below is absolutely convergent, conditionally convergent or divergent.
$$\sum_{n=1}^{+\infty}c_n\text{, where }c_n =\begin{cases}
-\frac{1}{n} ...
10
votes
2answers
181 views
Assigning values to divergent series
I have been looking at divergent series on wikipedia and other sources and it seems people give finite "values" to specific ones. I understand that these values sometimes reflect the algebraic ...
2
votes
2answers
374 views
Determine whether this series is absolutely convergent, conditionally convergent or divergent?
The series $ \sum_{n=1}^{\infty} \frac{(-1)^n n}{n^2 + 1} $; is it absolutely convergent, conditionally convergent or divergent?
This question is meant to be worth quite a few marks so although I ...
4
votes
2answers
92 views
comparison test - useful series to know
what are some useful series to know for the comparison test along with their conditions? I can think of the following:
p-series
geometric series
harmonic series
are there want other series that ...
2
votes
1answer
43 views
Series convergence help
If I have an increasing sequence of integers $$a_n$$ and $$
\sum_{n=1}^{\infty}\frac{\ln(a_n)}{a_n}$$
diverges, does $$\sum_{n=1}^{\infty}\frac{1}{a_n}$$ also diverge
14
votes
4answers
669 views
Why ${ \sum\limits_{n=1}^{\infty} \frac{1}{n} }$ is divergent , but ${ \sum\limits_{n=1}^{\infty} \frac{1}{n^2} }$ is convergent?
I don't understand why ${ \displaystyle \sum\limits_{n=1}^{\infty} \frac{1}{n} }$ is divergent, but ${ \displaystyle \sum\limits_{n=1}^{\infty} \frac{1}{n^2} }$ is convergent and its limit is equal to ...
16
votes
0answers
660 views
Prove that sum is finite
Let $j \in \mathbb{N}$. Set
$$
a_j^{(1)}=a_j:=\sum_{i=0}^j\frac{(-1)^{j-i}}{i!6^i(2(j-i)+1)!}
$$
and $a_j^{(l+1)}=\sum_{i=0}^ja_ia_{j-i}^{(l)}$.
Please help me to prove that the following sum is ...
3
votes
2answers
114 views
Can someone check if my proof is sufficient enough?
I've just started my undergraduate mathematics degree, and I have to say, proving things isn't very intuitive. I used to be very good at proofs in high school, but only when it comes to algebraic and ...
0
votes
5answers
80 views
Using the Comparison Test, prove that the infinite series of $(n^2+1)/(n^3+2)$ converges/diverges.
The conditions of the comparison sum state that if $0\le a_n\le b_n$
and if $b_n$ converges, then $a_n$ also converges
and if $a_n$ diverges, then $b_n$ also diverges.
I'm not sure how to go about ...
0
votes
1answer
178 views
Does this strange sum converge?
$$
\displaystyle \sum _{k=2}^{\infty } \sin \left(\frac{\pi k}{2}\right) \frac{B_k}{u^k}
$$
Notice that odd k terms add nothing to the summation because the Bernoulli term is zero and the sine term ...
0
votes
2answers
108 views
0
votes
1answer
203 views
Diverging to Positive and Negative Infinity
Say I have some sequence $\{a_n\}$ with one subsequence $\{a_{n_i}\} \longrightarrow \infty$ and another $\{a_{n_j}\} \longrightarrow -\infty$. In other words, the lim sup $a_n = \infty$ and lim inf ...
0
votes
1answer
1k views
How to determine if a series is convergent or divergent?
I have:
the series $\sum_{n=0}^{\infty} \frac{(2+\sin n)}{5^n} $.
I have split this up into $\sum_{n=0}^{\infty} \frac{2}{5^n} + \sum_{n=0}^{\infty} \frac{\sin n}{5^n}$. I know the first part is ...
1
vote
2answers
76 views
How to pick a comparison series to determine the convergence or divergence of a given series?
When testing to determine the convergence or divergence of series with positive terms, there's a common way by comparing them with other series which we already know converge or diverge.
My question ...
3
votes
2answers
84 views
real analysis converging subsequences
I am having trouble finding a diverging sequence in R whose subsequences of the even indexed terms.
5
votes
3answers
138 views
Is there a minimal diverging series?
Is there a function $f:\mathbb{N} \to \mathbb{R}^+$ s.t. its series $\Sigma_{i=0}^\infty f(n)$ diverges but the series for all function in $o(f)$ converge?
0
votes
2answers
51 views
Is there a diverging series in $o(1/n)$
Is there a function $f : \mathbb{N} \to \mathbb{R}^+$ in $o(1/n)$ s.t.
$\Sigma_{i=0}^\infty f(n)$ diverges?
6
votes
3answers
332 views
Is Fractal perimeter always infinite?
Looking for information on fractals through google I have read several time that one characteristic of fractals is :
finite area
infinite perimeter
Although I can feel the area is finite (at ...
3
votes
3answers
144 views
Does the series $\sum_{k=1}^\infty\left(\frac{1}{k}-\frac{1}{2^k}\right)$ diverge
Is there a handy way to tell if $\sum_{k=1}^\infty\left(\frac{1}{k}-\frac{1}{2^k}\right)$ diverges or not? I have a hunch that it diverges, since it looks like the sum is just $\zeta(1)-1=\infty$. But ...
13
votes
3answers
517 views
Convergence\Divergence of $\sum\limits_{n=1}^{\infty}\frac {1\cdot 3\cdots (2n-1)} {2\cdot 4\cdots (2n)}$
Prove convergence\divergence of the series:
$$\sum_{n=1}^{\infty}\dfrac {1\cdot 3\cdots (2n-1)} {2\cdot 4\cdots (2n)}$$
Here is what I have at the moment:
Method I
My first way uses a result that ...
10
votes
2answers
254 views
Summation of divergent series of Euler: $0!-1!+2!-3!+\cdots$
Consider the series
$$\sum\limits_{k=0}^\infty (-1)^kk!x^k\in\mathbb{R}[[x]]$$
Let $s_n(x)$ be partial sum. And let $\omega_{k,n}=(k!^2(n-k)!)^{-1}$. My question is:
Prove that ...
1
vote
0answers
56 views
series: can the result be zero for a continuous interval of its argument?
I'm considering the series
$$ f_c(x) = \sum_{k=c}^\infty \left( c^{k-1} \binom{k}{c} \cdot \prod_{j=1}^{k-1} (x-1/j) \right) $$
where the parameter $c \in \mathbb N ,c \gt 0$ and fixed for a certain ...
