Recurrence relations, convergence tests, identifying sequences
3
votes
1answer
177 views
How Can I Find The Value Of This Limit?
There is recurrence relation formula below.
$$\frac{1-c}{(n+1)^c}=(a_{n+1})^{1-c}-(a_{n})^{1-c}$$
$$c>0,\ c\neq1\ \ \ \ \ a_{n+1}>a_{n},\ a_1=1$$
Question. How can I find the limit below?
...
13
votes
1answer
525 views
Is there a constructive proof of this characterization of $\ell^2$?
I would like to revisit this question, which can be equivalently stated as:
Proposition. Let $(a_n)$ be a sequence of real (or complex) numbers such that $\sum a_n b_n$ converges for every $(b_n) ...
3
votes
1answer
95 views
Rate of change of $\mathbb{E}X_{\frac{2^n}{2},2^n}$ as $n$ increases?
I am trying to get an equation that will show the rate of change of the expected value of $\frac{2^n}{2}$th lowest of $2^n$ draws from $X$ as $n$ increases (where $n >1$). Let's call the order ...
4
votes
0answers
197 views
Equivalence of two sequences
I'm having some trouble showing that two things I really want to be the same are in fact the same. I want to show that these two sequences are, in fact, the same thing:
$$a_0=1,a_1=-1, ...
8
votes
1answer
212 views
An infinite series involving the Zeta Function
I am wondering if anyone knows how to evaluate either of the following sums in terms of known constants:
$$\sum_{k=2}^{\infty}-\frac{\zeta^{'}(k)}{\zeta(k)},$$
and
...
1
vote
1answer
105 views
Interpolation of a sequence of polynomials (viewed in terms of q-analogue powers)
I'm trying to find closed-form expressions for a sequence of coefficients, such that the index of the coefficient occurs as number such that I can later interpolate to fractional indexes as well. ...
6
votes
1answer
591 views
How to prove that this sequence converges?
I have some problems, with the convergence of this sequence defined recursively. It's clear that it's bounded. But is it convergent? How can I check for convergence?
$$ a_0 = a_1 = 1 $$
$$
a_n = ...
7
votes
1answer
197 views
A tricky series problem
The problem is the following:
If $\sum \limits_{n=1} ^{\infty} a_n$ converges, where $a_n$ are real numbers, then there exists $b_n \to \infty$ so that $\sum \limits_{n=1} ^{\infty} a_n b_n$ is ...
5
votes
3answers
290 views
A “fast” way to find the sum of the sequence $5,5.5,5.55,5.555,5.5555,\ldots $ (20 terms)
My initial approach is diving the whole sum by $9$ and taking the common $5$ out which gives $$\frac{5}{9}[(10-1)+(10-0.1)+(10-0.01)+\cdots + (10-10^{-19})]$$ after some algebra this could be reduced ...
4
votes
4answers
709 views
Infinite series expansion of $\sin (x)$
Are there any other ways to demonstrate that $$\sin(x)=\sum_{k=0}^{\infty}\frac{(-1)^kx^{1+2k}}{(1+2k)!}$$
without using the definition of Taylor series of complex exponentials, and similarly for ...
0
votes
0answers
93 views
Modular sequences
How might one show that if given that there exist some integer $x_0$ s.t. ${x_0}^2 \equiv c \pmod p$ for some integer $c$ that is not a square and not a multiple of $p$, then there exist $x_n$ s.t. ...
6
votes
1answer
511 views
Existence of a limit associated to an almost subadditive sequence
I want to prove that a sequence lives in a specific interval; I did prove that lives in a bigger interval, but not in the one I want.
Let $ a_n $ a sequence such that for any n,m
$$a_n + a_m -1 ...
0
votes
1answer
129 views
General term of sequence 3
Let be $$6,8,12,15,20,24,30,35,42,… $$ a sequence of natural numbers. Guessing the recurrence then using generating functions I can prove that general term of sequence is ...
4
votes
2answers
284 views
Does an infinite matrix exist with each row converging to 0 and each column to 1?
Is there an infinite matrix $A_{mn}$ such that $\lim\limits_{n \to \infty }A_{mn}=0 $ for every $m$ and $\lim\limits_{m \to \infty }A_{mn}=1 $ for every $n$ ?
Any clue as to how to start on this?
0
votes
0answers
129 views
Where can I find more information on the Hadamard Product (of Generating Functions)?
I've been messing around with generating functions, power series, and related series, and I've come across a simple method of using two "roots of unity filters" to calculate the Hadamard product. A ...
2
votes
2answers
198 views
this sequence converges?
I have some problems with this, because this sequence could converge to some point, or goes to infinity, only this two possibilities, and really I don´t know how to prove that no other possibility can ...
7
votes
2answers
742 views
Please help me to find the sum $\sum\limits_{n\geq1} \frac{\sin(nx)}{n^2}$
I must show that $\sum\limits_{n\geq1} \frac{\sin(nx)}{n^2}$ converges $\forall x \in {R}$. Then if $f(x)=\sum\limits_{n=1}^\infty f_n(x)$, I must prove that $f(x)$ is continuous for $x\in [0, \pi]$ ...
1
vote
5answers
278 views
Please help me to understand why $\lim_{n\to\infty } {nx(1-x^2)^n}=0$
This is the exercise:
"$f_{n}(x) = nx(1-x^2)^n, n \in {N}, f_{n}:[0,1] \to {R}. $ Find $ {f(x)=\lim\limits_{n\to\infty } {nx(1-x^2)^n}}$."
I know that $\forall x\in (0,1]$ $\Rightarrow (1-x^2) ...
6
votes
4answers
326 views
Please help me to prove the convergence of $\gamma_{n} = 1+\frac{1}{2}+\frac{1}{3}+…+\frac{1}{n}-\ln n$
Please help me to prove that $\gamma_{n} = 1+\frac{1}{2}+\frac{1}{3}+....+\frac{1}{n}-\ln n$ converge and it's limit $\gamma \in [0,1]$. Then, using $\gamma_{n}$ find the sum: ...
2
votes
1answer
145 views
Find the sum for $\sum_{n=1}^{\infty}(-1)^{(n+1)}\frac{2n+1}{3^n} $
I have been trying to find the sum $\sum_{n=1}^{\infty}(-1)^{(n+1)}\frac{2n+1}{3^n} $. After some calculation, I got here: $\frac{-6}{8}+\frac{1}{4}+8\sum\frac{k}{9^k}$. I know the result is ...
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 )}$$
$x$ and $y$ are arbitrary ...
7
votes
1answer
83 views
computing trigonometical
I have problem with the divergence of a Sum, I don´t know how criteria use when i has a trigonometrical function because, it´s not monotone, and I can´t prove that this series diverge with the usual ...
2
votes
3answers
646 views
How do I prove the divergence of this series?
How do I prove that $\displaystyle\sum_{n\geq 1}\frac {1}{\ln^2n}$ is a divergent series?
5
votes
2answers
381 views
Lagrange Inversion of power series for fractional exponents?
I understand how they obtained the inversion of sin(x) shown here, using the Lagrange Inversion Formula, and have even written a MATLAB script to solve the inversion when input and output exponents ...
1
vote
1answer
449 views
How can I find the constant of integration?
I'm attempting a novel approach to some tough integration problems. I'm using the idea of series expansions to help integrate. In other words, I will attempt to approximate integration by ...
3
votes
1answer
114 views
Limit of $\frac{(\sum_{1} ^ n a_j)^p}{n^{p - 1} \sum_{j = 1} ^ n a_j ^ p}$ where $\frac{\sum a_j}{n} \to \infty$
Let $a_1, a_2, ...$ be a sequence of positive numbers such that $\frac{\sum_{j = 1} ^ n a_j}{n} \to \infty$ as $n \to \infty$. What can we say about the behavior of $$\frac{(\sum_{j = 1} ^ n ...
4
votes
2answers
174 views
General term of sequence 2
An other sequence that arise in context of formula for number of partitions of number natural in parts non greater than 5 is
$81,123,167,229,295,381,473,587, 709, ...
0
votes
1answer
90 views
General term of sequence
In my work on number of partitions of natural numbers in parts non greater than 5 arise the sequence
$$775,1015,1285,1585,1915,2275,2665,3085,3535,4015,4525,5065,5635,6235,6865,7525,8215,8935,9685$$
I ...
1
vote
0answers
292 views
Finding a radius of convergence
Let $\sum_0^{\infty} a_n z^n$ have radius of convergence $R$ with $0< R< \infty$. Let $\alpha>0$. Find the radius of convergence of $\sum_0^{\infty} |a_n|^{\alpha} z^n$.
I tried to start ...
1
vote
2answers
238 views
For what values of $p\in\mathbb{R}$ does the series $\sum^{\infty}_{n=4}{\frac{1}{n\log (n)\log( \log(n))^p}}$ converge?
I'm having some trouble approaching this problem. "For what values of $p\in\mathbb{R}$ does the series $$\sum^{\infty}_{n=4}{\frac{1}{n\log (n)\log( \log(n))^p}}$$
converge?" For a fixed p, I could ...
3
votes
2answers
199 views
What is the reason for these jiggles when truncating infinite series?
Plotting the series $$\displaystyle y = \sum_{k} \frac{\sin kx }{k}$$
In the limit it would look like
Taking a finite number of terms, I want to understand what is the reason for the jiggling at ...
-2
votes
3answers
100 views
What are the values in this sequence [closed]
I have a trivia question to answer. What are the values (x,y,z) in this sequence
31,62,x,52,y,z,91
8
votes
3answers
477 views
What could be an intuitive explanation for $ \sum\limits_{k=1}^{\infty}\frac{1}{k2^k} = \log 2 $?
What could be an intuitive explanation for $\displaystyle \sum_{k=1}^{\infty}{\frac1{k\,2^k}} = \log 2$ ?
$\displaystyle \sum_{k=1}^{\infty}{\frac{1}{2^k}} = 1$ has a simple intuitive explanation ...
13
votes
2answers
625 views
“Pseudo-Cauchy” sequences: are they also Cauchy?
I tried to prove something but I could not, I don't know if it's true or not, but I did not found a counterexample.
Let $(a_n)$ be a sequence in a general metric space such that for any fixed $k ...
4
votes
2answers
156 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 $$\sum_{n = 1} ^ \infty \frac{a_n}{S_n^{1 ...
2
votes
0answers
199 views
How do I prove that the harmonic numbers are never integers? [duplicate]
Possible Duplicate:
Is there an elementary proof that $∑_{k=1}^n 1/k$ is never an integer?
I want to prove that the next number isn't an integer:
...
2
votes
1answer
222 views
partial sum of Basel problem related to series involving Beta function
I ran across a series and got to wondering how this is so.
We are all familiar with the famous $\displaystyle\sum_{k=1}^{\infty}\frac{1}{k^{2}}=\frac{{\pi}^{2}}{6}$
But, how can we show:
...
1
vote
0answers
168 views
Applications of Convergence of a series in Algorithms
We were introduced to testing the convergence of a series & calculating the point of convergence in the first maths of college curriculum. I wish to explore its usage in computer algorithms.
...
13
votes
3answers
300 views
A criterion for series convergence?
I have a conjecture regarding series convergence that feels like it would be a useful tool to me if I could prove it, but I have been unable to prove or disprove it.
Let $\sum a_n$ be a series of ...
5
votes
3answers
257 views
Perfect square sequence
In base 10, the sequence 49,4489,444889,... consists of all perfect squares.
Is this true for any other bases (greater than 10, of course)?
0
votes
4answers
164 views
Simple infinite sum identity
I'm looking for a concise way to show this:
$$\sum_{n=1}^{\infty}\frac{n}{10^n} = \sum_{n=1}^{\infty}\left(\left(\sum_{m=0}^{\infty}{10^{-m}}\right){10^{-n}}\right)$$
With this goal in mind:
...
4
votes
1answer
282 views
Weierstrass M-test, and $\sum_{n=1}^\infty e^{-nx}x^n$
How can I use the Weierstrass M-test to show uniform convergence of $\sum_{n=1}^\infty e^{-nx}x^n$ on $[0,\infty )$?
I can't find any bounding sequences. I've tried to analyse the convergence on a ...
9
votes
1answer
239 views
basic question--reasoning on alternating harmonic series
Can someone please tell me where this line of reasoning goes wrong?
False proof for the convergence of the alternating harmonic series:
Break the series $S = 1 - 1/2 + 1/3 - 1/4 + \dots$ into the ...
5
votes
2answers
238 views
What is $\sum\limits_{n=0}^{\infty} r^{an^2 + bn + c}$ ? or: is $0.0100100010000100001…$ transcendental?
The idea is a more convenient form for $N = 0.01001000100001000001...$ in base $r$, hopefully to show whether it is transcendental.
Sorry for brevity.
2
votes
6answers
638 views
Bernoulli's representation of Euler's number, i.e $e=\lim \limits_{x\to \infty} \left(1+\frac{1}{x}\right)^x $ [duplicate]
Possible Duplicates:
Finding the limit of $n/\sqrt[n]{n!}$
How come such different methods result in the same number, $e$?
I've seen this formula several thousand times: $$e=\lim_{x\to ...
4
votes
2answers
199 views
Convergence of $\sum \frac{1}{a_n}$ given convergence of $\sum a_n$
If we know that $\sum_{n=1}^{\infty}a_n$ converges,
what (if anything) can be known about
$\sum_{n=1}^{\infty}\frac{1}{a_n}$ ?
I understand that convergence means the summation adds up to a number ...
3
votes
4answers
96 views
Which test would be appropriate to use on this series to show convergence/divergence?
$$\sum_{n=1}^\infty \frac{(2n+1)^n}{n^{2n}}$$
I am struggling to come up with something. First thought root test but I don't think that will work since the roots aren't the same? I'm guessing maybe ...
20
votes
1answer
477 views
A fun Pascal-like triangle
A while back, inspired by Pascal, I put on some shackles and a thorny belt. Soon after, inspiration came pouring in, and I thought of the following triangle:
$$
\begin{array}{rcccccccccc}
& ...
1
vote
2answers
70 views
Transforming a sum of sequences
Let $f(i),i\in \mathbb N\, $ be a sequence of real or complex numbers then for natural numbers $m,n$ and $r$ holds sum transformation
...
0
votes
2answers
110 views
What is the mathematical formula representing this series?
I have the following sequences that form a relationship:
x: 1 2 3 4 5
y: 1 2 4 8 16
Where each number in the second sequence is twice the previous number.
Is there a formula that will give ...
