For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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0answers
12 views

Conjecture: $\lim_{x\to\infty} \frac{256}{163x}\sum_{n=1}^{\lfloor x\rceil} |\sin n|=1$

I conjecture: $$\lim_{x\to\infty} \frac{256}{163x}\sum_{n=1}^{\lfloor x\rceil} |\sin n|=1$$ Is this provable? If this is false, then can I have a function $f$ such that: $$\lim_{x\to\infty} ...
2
votes
2answers
36 views

Is it solvable to ask to find a recurrence relation for the following sequence 4,7,2,20,31,73,155,332,715,… ?

Is it solvable to ask to find a recurrence relation for the following sequence 4,7,2,20,31,73,155,332,715,... without any other information? If not, what would be the very least amount of information ...
0
votes
0answers
12 views

Function with both easy to find Fourier and Hermitian coefficient

I'm writing some notes on Spectral theory and I would like to make a simple example finding the generalized fourier coefficient of a function in respect of two different bases. I was thinking about ...
2
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1answer
47 views

Closed form of an integral $\int_0^{\pi/2} \ln^n (\sin x) \, dx$

Let $n \in \mathbb{N}$. May we have a closed form for the integral: $$\mathcal{J}=\int_0^{\pi/2} \ln^n (\sin x) \, {\rm d}x$$ One obvious approach would be to go through beta functions and ...
-2
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2answers
47 views

Solve this puzzle? [on hold]

Given a number, the answer is a power of $2$. Given $1.000$ the answer is $16384$. Given $5.000$ the answer is $131072$. Can someone find a function, so given any number we can get the answer?
-4
votes
2answers
136 views

Why is $\pi^2$ so close to $10$?

Noam Elkies explained why $\pi^2=9.8696...$ is so close to $10$ using an inequality on Euler's solution to Basel problem $$\frac{\pi^2}{6}=\sum_{k=0}^{\infty} \frac{1}{\left(k+1\right)^2}$$ to form ...
0
votes
1answer
110 views

Does this numerical series have any special name?

I don't have enough background to find the answer for this on my own, so I am posting it here with the hope to get some pointers. Assume a descending sequence of K numbers $\{n_1, ..., n_K\}$ where ...
0
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0answers
15 views

How many points among them should we take to ensure that some two of them are less than the distance $1/5$ apart?

We are given a fixed point on a circle of radius $1$, and going from this point along the circumference in the positive direction on curved distances $0,1,2,\ldots$ from it we obtain points with ...
1
vote
1answer
21 views

Show that $S^1$ is complete

Define: $S^1 = \{(x,y): x^2+y^2 = 1\}$ Then wish to show $S^1$ is complete Attempt: Let $(z_n)$ be a Cauchy sequence on $S^1$, $(z_n) = (x_n, y_n)$ Then $S^1$ is complete if $\forall ...
0
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1answer
25 views

Convergence of a series depending on a parameter

I have the following series $$\sum_{n=2}^{\infty} \frac{n}{(n-1)^2+\alpha 2^n}$$ I have to find for which $\alpha$ this series converges. I tried the ratio test but I get $\lim_{n \to \infty} ...
1
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0answers
17 views

On a “coincidence” of two sequences involving $a_n = {_2F_1}\left(\tfrac{1}{2},-n;\tfrac{3}{2};\tfrac{1}{2}\right)$

This was inspired by this post. Define, $$a_n = {_2F_1}\left(\tfrac{1}{2},-n;\tfrac{3}{2};\tfrac{1}{2}\right)$$ $$b_n = \sum_{k=0}^n \binom{\tfrac{1}{2}}{k}\big(-\tfrac{1}{2}\big)^k$$ where $_2F_1$ ...
-1
votes
2answers
7 views

Find Nth element where difference between elements is in AP [on hold]

Let I have a sequence as follows, $$a_1,a_2,a_3,a_4,a_5...$$ where $(a_2-a_1),(a_3-a_2),(a_4-a_3),... $ are in arithmetic progression. How can I find the Nth element ($a_N$) of the series ...
5
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0answers
40 views

Variation of the Kempner series

It is easy to argue that the Kempner series converges : $$ \sum\limits_{\substack{n \text{ : 9 is}\\\text{ not a digit of } n}} \frac{1}{n} < \infty$$ Let $E \subset \Bbb N_{>0}$ the subset ...
4
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1answer
28 views

Proving that the Calkin-Wilf tree enumerates the rationals.

The Calkin-Wilf tree is an infinite undirected graph (tree) which is constructed as follows: starting from the root at $\frac{1}{1}$, each node $\frac{a}{b}$ has two children: a left child ...
3
votes
1answer
455 views

Uniform convergence of $f_n = (n^a x^2)/(n^2 +x^3)$

My question is, if you have the sequence $$f_n = \frac{n^\alpha x^2}{n^2 +x^3}$$ on $[0, \infty)$, for values of a for $0<\alpha<2$ does the sequence uniformly converge? I guess another way to ...
0
votes
0answers
14 views

Solving summation with incomplete gamma function

I solved an indefinite integral that gave me the following result: \begin{equation} f(u) = -\sum_{k=0}^{n}\binom{n}{k}\frac{\mathrm{sgn}(u)^{k+1}\, b^{n-k}}{2\, a^{\frac{k+1}{2}}}\, ...
0
votes
1answer
39 views

Limit of a sequence with binomial coefficient. Can I use Stirling?

I was trying to solve this limit: $\lim_\limits{n\to \infty} \binom {3n}{n}^{1/n} $ I solved it with Cesaro theorem: $\lim_\limits{n\to \infty} \binom {3n}{n}^{1/n} $= $\lim_\limits{n\to \infty} ...
0
votes
0answers
22 views

$\lim\limits_{K\to\infty} [n,K]$.

I've seen the following notation sometimes $$\lim\limits_{K\to\infty} [n,K]$$ And some people claim this is equal to $[n,\infty)$. They say, well $K$ keeps getting larger so for whatever $x\ge n$, we ...
4
votes
3answers
333 views

Is there any partial sums of harmonic series that is integer?

is there any partial sums of harmonic series that add up to an integer? partial sums not as trivial as the first term only i.e. 1, or the powers of 2 i.e the infinite geometric series for 2. This ...
0
votes
1answer
68 views
+50

all but one sub-strings within a cyclic string

over $GF(q)$ where $q\in\mathbb{N}$, we build a string of size $q^n-1$. now, how can I show that it is always possible to construct that string so it contains all sub-strings of size $n$ exactly once, ...
2
votes
2answers
51 views

Show $\forall \delta > 0, \exists n \in \mathbb{N}$ such that $\frac{1}{n} < \delta$

The question is in the title, but I have no idea how to solve it, so a few hints would be appreciated, thanks.
0
votes
1answer
34 views

Why does $x_n^2$ converge to $x^2$ if $x_n$ converges to $x$?

Let $x_n$ be a convergent sequence converging to $x$ Then claim $x_n^2$ converges to $x^2$ I wish to use the definition to show this is the case. Recall $x_n \to x$ iff $\forall \epsilon > 0, ...
1
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5answers
37 views

Prove $a_t \rightarrow x$ using the Betweenness Property

Prove that for any $x \in \Bbb R$ there is a strictly increasing sequence ($a_t$) in $\Bbb Q$ such that ($a_t$) converges to $x$, (i.e. $a_t \rightarrow x$) I want to prove this using the Betweenness ...
5
votes
1answer
33 views

$\lim_{n \to \infty}(\frac{a_n}{\sqrt{a_n^2+1}})=\frac{1}{2}$ - show that $a_n$ is convergent sequence

Problem: Show that $a_n$ is convergent sequence and find a limit of $a_n$. $$\lim_{n \to \infty}(\frac{a_n}{\sqrt{a_n^2+1}})=\frac{1}{2}$$ I tried to look at this as normal limit problem so I wrote ...
8
votes
2answers
78 views

$p = \sqrt{1+\sqrt{1+\sqrt{1 + \cdots}}}$; $\sum_{k=2}^{\infty}{\dfrac{\lfloor p^k \rceil}{2^k}} = ? $

Let $p = \sqrt{1+\sqrt{1+\sqrt{1 + \cdots}}}$ The sum $$\sum_{k=2}^{\infty}{\dfrac{\lfloor p^k \rceil}{2^k}}$$ Can be expressed as $\frac{a}{b}$. Where $\lfloor \cdot \rceil$ denotes the ...
1
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0answers
15 views

Infinite series, continued fractions, nested radicals and such - is there a general recursive algorithm theory?

The nature of infinity and irrational numbers (among other things) always gave me trouble. But recently I learned to think about infinite sequences in terms of recursive algorithms and their time ...
20
votes
4answers
917 views

Elementary Proof of Euler's Sine Expansion $\prod_{k=1}^\infty\left({1-\frac{x^2}{k^2\pi^2}}\right)$

I've been looking at proofs of Euler's sine expansion, that is $$\frac{\sin x}{x}=\prod_{k=1}^\infty\left({1-\frac{x^2}{k^2\pi^2}}\right)$$ All the proofs seem to rely on complex analysis and ...
107
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9answers
11k views

Is there an elementary proof that $\sum \limits_{k=1}^n \frac1k$ is never an integer?

If $n>1$ is an integer, then $\sum \limits_{k=1}^n \frac1k$ is not an integer. If you know Bertrand's Postulate, then you know there must be a prime $p$ between $n/2$ and $n$, so $\frac 1p$ ...
-1
votes
0answers
17 views

Solution to recursive equation

what will be the form of solution for this kind of recurrence equation? $$P_{n+1} + \dfrac{2n P_n}{x} - P_{n-1} = 0$$ $x$ is a constant. Will a guess solution of form $\lambda^n$ work? I need to ...
0
votes
1answer
47 views

Convert Riemann sum to definite integral: $\sum_{i = 1}^n \frac{n}{n^2 + i^2}$

I am having trouble with this problem. Basically, I am given a Riemann sum and I have to rearrange it so that I can deduce the definite integral that it is equivalent to. Thank you. $$\lim_{n \to ...
0
votes
2answers
55 views

Finding an expression for the sum of n tems of the series $1^2 + 2^2 + 3^2 + … + n^2$ [duplicate]

Possible Duplicate: why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$ I know that if you have a non-arithmetic or geometric progression, you can find a sum $S$ of a ...
1
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1answer
44 views

If $\sum\limits_n \sqrt{\mathrm{Var}(X_n)}$ converges, then $\sum\limits_n (X_n - E[X_n])$ converges in $L^2$

I have seen in a paper to claim the following: If $\sum\limits_n \sqrt{\mathrm{Var}(X_n)} < \infty$, then $\sum\limits_n (X_n - E[X_n])$ converges in $L^2$, for any sequence of random ...
2
votes
2answers
48 views

Comparing a Factorial and a Perfect Power

Let us define the following recurrence relations as so. $$a_1=6, a_{n+1}=a_n!$$ $$b_1=6, b_{n+1}=6^{b_n}$$ So, which of the following is larger? $a_{b_2}$ or $b_{a_2}$? To clarify, I am trying to ...
0
votes
1answer
15 views

Why the dual to $c_0$ is $l^1$ and the space of sequences with bounded partial sums?

The dual to $c_0$ is $l^1$, but if $\{x_n\}_{n\in\mathbb{N}}\in c_0$, than according to Dirichlet's test $\sum_{n\in\mathbb{N}}(-1)^nx_n$ converges. But $\{(-1)^n\}_{n\in\mathbb{N}}\notin l^1$. So why ...
0
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2answers
24 views

Showing the set $A = \{ x \in l_1 : |x_n| \leq 1/n^2 ,\forall n\}$ is closed

Showing the set $A = \{ x \in l_1 : |x_n| \leq 1/n^2 ,\forall n \}$ is closed. I had to show it is compact, and I am done showing it is relatively compact, but now I am stuck showing it is closed. ...
0
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0answers
14 views

How could I prove that equivalence on limits of sequences?

$T$ is a positive real number $\lambda_k$ is a family of positive reals number that $\to \infty$ with $k$ $f(s)$ is a function that is a $o(s)$ and $0<f(s)<s$ (these properties may be ...
0
votes
2answers
21 views

Calculating limit of a series of series

Given is the following series I want to calculate the limit for $n \to \infty$. I already recognised the geometric series $\sum_{i=2}^n a^{i} = \frac{1}{1-a}$ for $a=e^\rho$ (since rho is ...
6
votes
0answers
53 views

A closed form for the following Series

I was computing some calculations, when I got stuck about a possible closed form for this series: $$S = \sum_{k = 2}^{N}\ \frac{k!}{k^k - k!}$$ I proved by hands that it's absolutely convergent by ...
1
vote
1answer
77 views

Why does $\sum\limits_{n=0}^{+\infty} z^n=\frac{1}{1-z}?$

Having $f(z)=\sum\limits_{n=0}^{+\infty} \frac{1}{n!}z^n$ I had to find what $\sum\limits_{n=0}^{+\infty} \frac{1}{n!}z^n\sum\limits_{n=0}^{+\infty} \frac{D_n}{n!}z^n=\sum\limits_{n=0}^{+\infty} ...
2
votes
1answer
45 views

If a sequence of functionals converges weakly then it is bounded.

Let $f_k, f \in L^{\infty}(R)$ and $f_k \overset * \to f$ in $L^{\infty}(R)$. Is $f_k$ a bounded sequence in $L^{\infty}(R^n)$? (Definition: if $(v_n)$ is a sequence in $V = X^*$, we say that $v_n ...
-1
votes
2answers
41 views

$\lim_{n\to\infty}{a_{n}} =\lim_{n\to\infty}{b_{n}}$ iff $\lim_{n\to\infty}{(a_{n}-b_{n})=0}$ [on hold]

I need to proof or disproof that: $$\lim_{n\to\infty}{a_{n}} =\lim_{n\to\infty}{b_{n}}$$if and only if $$\lim_{n\to\infty}{(a_{n}-b_{n})=0}$$
0
votes
0answers
21 views

What kind of information does the derivative of a sequence of functions give us?

When we derivate a function we know for example when then function increases and drecreases but when I derivate a sequence of functions I don't how to interpret the derivative since it is a collection ...
23
votes
4answers
833 views

How close can $\sum_{k=1}^n \sqrt{k}$ be to an integer?

How close can $S(n) = \sum_{k=1}^n \sqrt{k}$ be to an integer? Is there some $f(n)$ such that, if $I(x)$ is the closest integer to $x$, then $|S(n)-I(S(n))|\ge f(n)$ (such as $1/n^2$, $e^{-n}$, ...). ...
1
vote
3answers
60 views

Solve the following sequence problem

Let a sequence be defined as $$a_n=\lim_ {x \to 0}{1-\cos (x)\cos(2x).....\cos (nx)\over x^2}$$ a)prove that the given sequence is monotonic and that it is not bounded above. b)calculate $$\lim_{n \to ...
1
vote
2answers
24 views

Alternative version of definition of convergence

Knowing the definition of convergence of a sequence in $\Bbb R$: ($x_n$) converges to x iff $\forall \epsilon \gt 0$ $\exists N$ such that for every $n\gt N$, $d(x_n,x)\lt \epsilon$. Consider a new ...
0
votes
2answers
20 views

How do I use the ratio test to determine convergence or divergence in this problem?

I have the problem: $$a_{n} = \frac{e^{n+5}}{\sqrt{n+7}(n+3)!}$$ I am told to use the ratio test to determine convergence or divergence (or the test could be inconclusive). So I take the limit: ...
0
votes
0answers
6 views

Lagarias and Robin theorems versus multiplicative property

If I use for example Robin's theorem, see here in the section Growth of arithmetic functions, or Lagarias equivalence, see (5) here has sense ask us what is the more sharp inequality for ...
11
votes
4answers
504 views

Finding the power series for $y$ where $y + \sin(y) = x$

What do you do to find the power series for an inverse relationship such as for $y$ in $y + \sin(y) = x$? I'm not sure where to begin. (Similarly, the Lambert $W$ function has such a power series ...
0
votes
1answer
23 views

Why is the following reverse triangle inequality true for given series?

I wish to show that for $(a_k)$ a sequence of numbers, $a_k \in \mathbb{R}$ then claim : $|\sum\limits_{k = n+1}^m a_k | \leq ||\sum\limits_{k = n+1}^\infty a_k| - |\sum\limits_{k = ...
-1
votes
0answers
16 views

Express $\prod \frac{a_i}{x+b_i}$ in terms of known functions [on hold]

I am interested if there is a way to express the finite product $$ \prod_{i=1}^{S} \frac{a_i}{x+b_i} $$ in terms of known functions like Gamma, Beta, etc.