For questions about recurrence relations, convergence tests, and identifying sequences

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4
votes
1answer
56 views

Does this sequence consist of squares of integers?

Question: let sequence $\{x_{n}\}$ such $$x_{0}=0,x_{1}=1,x_{2}=0,x_{3}=1$$ and such $$x_{n+3}=\dfrac{(n^2+n+1)(n+1)}{n}x_{n+2}+(n^2+n+1)x_{n+1}-\dfrac{n+1}{n}x_{n}$$ show that:$ ...
17
votes
0answers
161 views
+50

Evaluating the sum $\lim_{n\to \infty}\sqrt[2]{2+\sqrt[3]{2+\sqrt[4]{2+\cdots+\sqrt[n]{2}}}}$

The following nested radical $$\lim_{n\to \infty}\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}$$ is known to converge to 2. We can consider a similar nested radical where the degree of the radicals increases: ...
28
votes
5answers
4k views

Help me solve my father's riddle and get my book back

My father is a mathteacher and as such he regards asking tricky questions and playing mathematical pranks on me once in a while as part of his parental duty. So today before leaving home he sneaked ...
4
votes
1answer
30 views

How is $ \sum_{n=1}^{\infty}\left(\psi(\alpha n)-\log(\alpha n)+\frac{1}{2\alpha n}\right)$ when $\alpha$ is great?

Let $\psi := \Gamma'/\Gamma$ denote the digamma function. Could you find, as $\alpha$ tends to $+\infty$, an equivalent term for the following series? $$ \sum_{n=1}^{\infty} \left( \psi (\alpha ...
2
votes
2answers
77 views

A problem on nested radicals

Find the value of $x$ for all $a>b^2$ if: $$\large x=\sqrt{a-b\sqrt{a+b\sqrt{a-b{\sqrt{a+b.......}}}}}$$ My attempt $$\large x=\sqrt{a-b\sqrt{(a+b)x}}$$ $$\large x^4=(a-b)^2(a+b)x$$ $$\large ...
5
votes
2answers
95 views

From the series $\sum_{n=1}^{+ \infty} \left(H_{n}-\ln n-\gamma -\frac{1}{2n}\right)$ to $\zeta(\frac{1}{2}+it)$

Here is a pretty series $$ \displaystyle \sum_{n=1}^{+ \infty} \left(H_{n}-\ln n-\gamma -\frac{1}{2n}\right)=\frac{1}{2} \left(1-\ln (2\pi)+\gamma\right) \quad (*) $$ where $H_{n}:=\sum_{1}^{n} ...
1
vote
0answers
57 views

Successive ratios of a sequence, is this limit true?

The natural numbers $1,2,3,4,5....$ can be calculated as the row sums of the triangle $T(n,k)$ equal to $1$ if $n \geq k$ and $0$ otherwise: $$\displaystyle T = \left(\begin{matrix} ...
0
votes
2answers
44 views

Supremum of $f_1(x)=1$ and $f_2(x)=x$

I'm trying to understand the supremem of a sequence of functions so I came up with a trivial case as follows - Let $(f_n(x))$ be a sequence of functions with $n$ having a value of either $1$ or $2$. ...
25
votes
1answer
999 views

Show that $\int_{0}^{\pi/2}\frac {\log^2\sin x\log^2\cos x}{\cos x\sin x}\mathrm{d}x=\frac14\left( 2\zeta (5)-\zeta(2)\zeta (3)\right)$

Show that : $$ \int_{0}^{\pi/2} {\ln^{2}\left(\vphantom{\large A}\cos\left(x\right)\right) \ln^{2}\left(\vphantom{\large A}\sin\left(x\right)\right) \over ...
1
vote
1answer
47 views

Convergence of the series $\sum\limits_{n\geqslant1}(2-x)(2-x^{1/2})(2-x^{1/3})\cdots(2-x^{1/n})$

If $x >0$, consider the series $\sum\limits_{n=1}^\infty a_n$, where $$a_{n}=(2-x)(2-x^{1/2})(2-x^{1/3})\cdots(2-x^{1/n}).$$ Is it convergent? This question is in my textbook. I don't know how to ...
2
votes
2answers
68 views

A transcendental number from the diophantine equation $x+2y+3z=n$

Let $\displaystyle n=1,2,3,\cdots.$ We denote by $D_n$ the number of non-negative integer solutions of the diophantine equation $$x+2y+3z=n$$ Prove that $$ \sum_{n=0}^{\infty} ...
1
vote
2answers
35 views

Intuition for sequences of functions?

A sequence $(a_n)$ of real numbers can be thought of as a function that maps $\mathbb{N}$ to $\mathbb{R}$. The supremum of this sequence, if it exists, will be some $k \in \mathbb{R}$. A regular ...
0
votes
1answer
12 views

Increasing numbers of interations, patterns

Write expression for e to the power of i with increasing numbers of interations, simplifying wherever possible, comment on patterns discovered throughout the equation. Help would be appreciated
1
vote
1answer
29 views

How to find the Direct Discrete Laplace Transform of ${2n \choose n}$

Some time ago I developed a discrete version of the Laplace transform for the purpose of calculating sums and solve finite difference equations with constant coefficients. The notes below are a ...
3
votes
0answers
20 views

Convergence of sum of antiderivative and derivative

This question is inspired by this question: Solutions for $ \frac{dy}{dx}=y $?. It makes me wonder if there are any function where the sum of all antiderivative and derivative converges. The ...
1
vote
0answers
14 views

Evaluation of a Hankel-like determinant

I consider the following determinant (Hankel-like?) $$ [f_1,f_2,...,f_n]:=\begin{vmatrix} f_1 & f_2 & \cdots & f_{n-1} & f_n\\ n-1 & f_1 & \cdots & f_{n-2}& f_{n-1}\\ 0 ...
2
votes
2answers
27 views

Evaluation of Indefinite Integral resulting in Hypergeometric Function

I am attempting to derive the result: $$ \int \left(1+x^n\right)^{-1/m}dx= x\,_2F_1\left(\frac 1m,\frac 1n;1+\frac 1n;-x^n\right)$$ First, I start off with the binomial expansion of the integrand to ...
0
votes
0answers
23 views

How to count the number of square arrangements?

I need to know the number of combinations that can be made with the linking previous blocks horizontally and vertically (not diagonally) for n blocks. All blocks must touch each other horizontally or ...
4
votes
6answers
182 views

Find the $n$th term of $1, 2, 5, 10, 13, 26, 29, …$

How would you find the $n$th term of a sequence like this? $1, 2, 5, 10, 13, 26, 29, ...$ I see the sequence has a group of three terms it repeats: Double first term to get second term, add three to ...
0
votes
0answers
36 views

Proving pointwise convergence of Fourier series [on hold]

I'm trying to prove the pointwise convergence of a $2\pi$-periodic function $f$ to its Fourier expansion. The proof on my lecture notes stops at this formula: $$f(t)-P_{N,f}=\frac 1 {2 \pi} ...
0
votes
2answers
43 views

double summation problem $\sum^5_{i=1}i \times \sum^5_{j=1}j =…$ please check

(I) $\sum^5_{i=1}i \times \sum^5_{j=1}j = 1 \times (1) +1 \times (2) + \cdots +1\times (5) +2\times (1)+2\times (2) +\cdots + 2\times (5) + 3\times (1) + 3\times (2) + \cdots +3\times (5) + 4\times ...
3
votes
1answer
56 views

$1/k=\sum_{n=1}^\infty a_n^k$ for all k

Suppose $1/k=\sum_{n=1}^\infty a_n^k$ for all integers $k>1$, what are all the sequences of positive real numbers $a_i$ that satisfies this set of equations?
2
votes
2answers
21 views

Proving arithmetic series by induction

How do I prove this statement by the method of induction: $$ \sum_{r=1}^n [d + (r - 1)d] = \frac{n}{2}[2a + (n - 1)d] $$ I know that $d + (r - 1)d$ stands for $u_n$ in an arithmetic series, and the ...
1
vote
1answer
46 views

Closed form for the recursion $\displaystyle u_n=\sum_{k=0}^{n-1} u_ku_{n-1-k}$

I was completing a computer science problem when the following recursion popped up: $u_0=1$ $\displaystyle u_n=\sum_{k=0}^{n-1} u_ku_{n-1-k}$ Is there a closed form for this recursion ? I ...
1
vote
2answers
55 views

Convergence of Sequence $a_n=1+\frac{1}{4}+\frac{1}{7}+\ldots+\frac{1}{3n-2}$

Apply Cauchy's principle of convergence to prove that the sequence $\langle a_n\rangle$ defined by $$a_n=1+\frac{1}{4}+\frac{1}{7}+\ldots+\frac{1}{3n-2}$$ is not convergent My attempt : consider, ...
215
votes
21answers
19k views

Different methods to compute $\sum\limits_{n=1}^\infty \frac{1}{n^2}$

As I have heard people did not trust Euler when he first discovered the formula $$\zeta(2)=\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler and he gave other proofs. I ...
0
votes
2answers
47 views

Prove that no polynomial function represents a given sequence

I recently encountered the following question: How to find the $n$th term of the sequence $2,3,6,7,14,15,30,\dots$? I replied to that post, and gave the following answer: $S_n = ...
2
votes
4answers
44 views

Prove a sequence does not tend to zero

Prove that the sequence $\dfrac{a^n}{2^nn^2}$ where $a>2$ does not tend to zero. I thought about writing $a=2+{\epsilon}$ then using binomial expansion which is valid for ${\epsilon}<1$ but ...
0
votes
2answers
42 views

Prove that $\sum_{k = 0}^{r}\binom{n + k}{ k} = \binom{n+r+1}{ r}$ whenever $n$ and $r$ are positive integers.

Question: Prove that $\displaystyle\sum_{k = 0}^{r}\binom{n + k}{ k} = \binom{n+r+1}{r}$ whenever $n$ and $r$ are positive integers. a.) using combinatorial argument. b.) using Pascal's identity. ...
1
vote
1answer
66 views

Alternate series [duplicate]

The alternate series $S=\displaystyle \sum_{k=2}^{\infty} \frac{(-1)^n}{\sqrt{n}+(-1)^n} $ converges? $S$ is absolutely convergent?
-1
votes
0answers
41 views

Infinite radius of convergence

If the root test limit tends to infinity, is that sufficient to say we have an infinite radius of convergence? Thanks
3
votes
1answer
124 views

How to prove that $\sum_{k=0}^\infty \binom{x}{x-k}\cdot\binom{x}{k-x} = 1$?

How to prove this: $$\sum_{k=0}^\infty \binom{x}{x-k}\cdot\binom{x}{k-x} = 1$$ For all $x\in\mathbb R_{\ge0}$ and with $\binom{x}{r}=\frac{\Gamma(x+1)}{\Gamma(r+1)\cdot\Gamma(x-r+1)}$ It is obviously ...
0
votes
1answer
41 views

How do I solve the following limit?

The solution to this limit should be 1, but I don't know how to solve it. I suspect I should rewrite the sequence but it's not geometrical or arithmetic as far as I can see. $\lim _{x\to \infty ...
2
votes
1answer
73 views

Power series difficulty

How would I find the region of convergence of the series of $\frac{1}{n^3}(\frac{z+1}{z-1})^n$. I thought about rewriting $\frac{z+1}{z-1}$ as $\frac{2}{z-1}+1$ but I don't think that helps. Thanks
0
votes
2answers
25 views

Radius of convergence query

Find the radius of convergence of the series of $\frac{2^n(4z-8)^n}{n}$ My answer: $(4z-8)^n=4^n(z-2)^n=2^{2n}(z-2)^n$. Let $c_{n}=\frac{2^{3n}}{n}$. Then $\frac{c_{n}}{c_{n+1}}=\frac{n+1}{2n}$ so ...
4
votes
3answers
107 views

Prove $ne^{-n}$ converges to zero

How would I prove that $ne^{-n}$ converges to zero? I've tried $ne^{-n}<{\epsilon}$ and then logging both sides but no further progress could be made. Thanks
1
vote
1answer
37 views

Show that {$a_n$} is convergent and find sup{$a_n| n \in Z_+ $}

$a_1 = 1$ and $ a_{n+1} = \frac{4+3a_n}{3+2a_n} ; \forall n \in Z_+$ Show that {$a_n$} is convergent, find its limit and find sup{$a_n| n \in Z_+ $} if exists. I found the limit as follows - ...
9
votes
2answers
1k views

Multiples of an irrational number forming a dense subset

Say you picked your favorite irrational number $q$ and looking at $S = \{nq: n\in \mathbb{Z} \}$ in $\mathbb{R}$, you chopped off everything but the decimal of $nq$, leaving you with a number in ...
0
votes
0answers
41 views

Another proof of the Dirichlet's test

My teacher said, that the Dirichlet's test was equivalent to the lemma as follows, and the lemma could be proved with an estimate without using Abel's summation formula. He expected me to complete the ...
3
votes
2answers
24 views

radius of convergence when root test fails

I'm stuck on this problem: Find the radius of convergence of $\sum \limits_{n=1}^\infty \frac{x^n}{((2+(-1)^n)^n} $ An attempt: From the root test, it seems $L$ does not exist: $$L= \lim_{n ...
1
vote
2answers
45 views

Find a closed form for the generating function for this sequence

The sequence: $0, 0, 0, 1, 1, 1, 1, 1, 1, \ldots$ The book gives the answer of $\frac{x^3}{1-x}$ but I'm not sure how to get this answer. I understand the generating function of this sequence will be ...
3
votes
2answers
50 views

Infinite series involving sum of divisors function

Is there much known about infinite series of the form $\sum_{n=1}^{\infty}\frac{\sigma_{1}(n)}{n}q^{n}$ where $\sigma_{1}(n)$ is the sum of divisors function. I am particularly interested in ...
1
vote
2answers
34 views

A trouble with the discrete product topology

Consider a finite set $S=\{1,\ldots,n\}$ with the discrete topology, and moreover construct the product topological space $S^\mathbb N$ with the product topology. $S^\mathbb N$ is made by all the ...
0
votes
1answer
75 views

Need help with a integral

I was evaluating $$ \int_{0}^{^\pi/_2}x\ln\left(\vphantom{\large A}\cos\left(x\right)\right)\,{\rm d}x $$ I like to try with the fourier series $$ \int_{0}^{^\pi/_2} \left[\,\,\sum_{k = 1}^{\infty} ...
4
votes
3answers
346 views

I believe this is a Taylor series. How do I approach it, and what formulas can I use to solve this type of problem?

Suppose that $|x| < 1$. Find the sum of the series $$2x - 4x^3 + 6x^5 - 8x^7 + \cdots$$ I'm not looking for an answer. I want to know how to appropriately solve such a question though.
6
votes
4answers
371 views

A generalized integral need help

I was thinking this integral : $$I(\lambda)=\int_0^{\infty}\frac{\ln ^2x}{x^2+\lambda x+\lambda ^2}\text{d}x$$ What I do is use a Reciprocal subsitution, easy to show that: ...
1
vote
1answer
74 views

Series involving harmonic numbers

Denote by $H_i$ the $i$-th harmonic number. I conjecture that $$\lim\limits_{n\rightarrow \infty} H_n^2 - 2\sum\limits_{i=1}^n \frac{H_i}{i}$$ exists. I have no proof for this. I only have a vague ...
0
votes
0answers
30 views

Convergence of a series using limit test

As per my understanding: When we take $\lim_{n \rightarrow \infty}$ of a function, it should approach a finite number, it converge. And if the opposite is true, it diverges. Now, the test for ...
23
votes
5answers
641 views

About a harmonic series problem : How to prove that $\sum_{n=1}^{\infty}\frac{H_n}{n^3}=\frac{\pi^4}{72}$

How to prove that $$\sum_{n=1}^{\infty}\frac{H_n}{n^3}=\frac{\pi^4}{72}$$ $H_n$ is the n th harmonic number
14
votes
3answers
450 views

A series involves harmonic number

How do we get a closed form for $$\sum_{n=1}^\infty\frac{H_n}{(2n+1)^2}$$