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

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2
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0answers
12 views

Measuring sums of complex alternating series

Suppose we have functions $$f(x) = \sqrt{x}, \space g(f) = \frac{df}{dx}+\frac{d^2f}{dx^2}+\frac{d^3f}{dx^3}\space ...$$ Applying function f(x) to g(f) we get: $$g(f(x))=\frac{1}{2}x^{-\frac{1}{2}} - ...
2
votes
0answers
57 views

An arithmetic sequence whose members do not contain the digit ‘9’

There is an arithmetic progression made of natural numbers only; none of them contains the digit $9$. Prove that the number such an arithmetic progression has no more than $72$ terms.
2
votes
2answers
36 views

is there a generating function for $H_{2n}$?

I have been wondering if anyone knows if there is a generating function for harmonic series of the form $H_{2n}$?. That is, we are familiar with ...
4
votes
1answer
43 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 ...
4
votes
1answer
71 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:$ ...
0
votes
2answers
48 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 $2$. ...
2
votes
2answers
80 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 ...
1
vote
1answer
49 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 ...
1
vote
2answers
36 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
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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
3
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0answers
21 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
15 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 ...
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 ...
2
votes
2answers
78 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} ...
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} ...
31
votes
5answers
5k 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 ...
3
votes
1answer
58 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?
1
vote
0answers
58 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} ...
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 ...
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 ...
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 = ...
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, ...
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 ...
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?
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. ...
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0answers
42 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
2
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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 ...
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 ...
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 - ...
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
47 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 ...
1
vote
2answers
35 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 ...
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 ...
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 ...
0
votes
1answer
18 views

Mixing arithmetic and geometric progressions

I'm having trouble blending two different types of progressions: The fourth, eighth and fourteenth terms of an A.P., common difference 0.5, are in geometric progression. Find the first term of the ...
5
votes
2answers
96 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} ...
4
votes
6answers
183 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 ...
-1
votes
0answers
47 views

Sum of arctan with recurrence relation [on hold]

Let $\{F_n\}_{n=1}^\infty$ be a sequence satisfies recurrence relation $F_n=15F_{n-1}-15F_{n-2}+F_{n-3}$ for all $n\geq4$, where $F_1=4$, $F_2=64$, $F_3=900$. Find the value of ...
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 ...
0
votes
1answer
19 views

Geometric Progressions: Finding the number of terms that will double the first term

If the value of an article is assumed to increase annually by 5% of its value at the beginning of the year, after how many years will its value double. Here is what I've done so far: Value at ...
0
votes
1answer
6 views

Finding Possible Values of GP Common ratio (r)

r is the common ration of a GP (r is not equal to 1) and the sum of the first 4 terms is 5 times the sum of the first 2 terms. Find the possible values of r. How do I solve this one? Thanks.
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0answers
21 views

What are the practical applications of quadratic and cubic squence? [on hold]

I would like to know, whether there are any practical uses for quadratic and cubic sequences?
2
votes
2answers
37 views

Integral to measure error within 10^-8

If someone could give me background on HOW to solve this problem, NOT THE ANSWER, that would be appreciated. I would love to know how to approach this problem in the most efficient and universal way. ...
4
votes
2answers
59 views

An asymptotic term for a finite sum involving Stirling numbers

The question is a by-product at the end of this post. The following asymptotic term will ensure the convergence of some series. $$ \frac{1}{n!} \sum_{k = 1 }^{n } \frac{{n \brack k}}{k+1} = ...
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.
1
vote
1answer
26 views

Property of Conditionally Convergent series

If $ \sum a_n$ be an conditionally convergent series.For any real number R, is it true that there exists a sequence$\{b_n\}$ where each $b_i=1 $ or $-1$ such that $\sum a_nb_n$ converges to R?
4
votes
3answers
108 views

Comparison test for sequences?

Let $a_n, b_n$ such that for sufficiently large $n$: $ a_n \le b_n$. Can we deduce that: $\lim_{n\to\infty}a_n = \infty \implies \lim_{n\to\infty}b_n = \infty$ $\lim_{n\to\infty}b_n = L ...