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

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3
votes
1answer
62 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
0answers
80 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
29 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
47 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
53 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
56 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
51 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
69 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
44 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
votes
0answers
46 views

Infinite radius of convergence [on hold]

If the root test limit tends to infinity, is that sufficient to say we have an infinite radius of convergence? Thanks
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
26 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
44 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
109 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
26 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
48 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
39 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
31 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
114 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
190 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
49 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
46 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.
-3
votes
0answers
21 views

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

I would like to know, whether there are any practical uses for quadratic and cubic sequences?
2
votes
2answers
38 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
60 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
350 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
28 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
109 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 ...
0
votes
2answers
122 views

Fibonacci series, which is most pure mathematically? [closed]

There are various methods of generating the Fibonacci series. I'm going to list 3 of them in this question. Method 1: f[n] = f[n - 1] + f[n - 2] Method 2: (more ...
1
vote
2answers
73 views

prove ${a_n} = \root n \of {n!} $ is monotonically increasing to $\infty$

Prove ${a_n} = \root n \of {n!} $ is monotonically increasing to $\infty$ I already showed that $a_n$ diverges to infinity like this: I used to the lemma which says that if ...
0
votes
2answers
24 views

Associative property for series

Are those equation always valid: $$\sum a_n + b_n = \sum a_n + \sum b_n$$ $$\sum_{k=1}^n(a_{k+1}+a_k)-\sum_{k=1}^na_k=\sum_{k=1}^na_{k+1}$$
0
votes
1answer
34 views

Prove that the elements of these two sequences are not null

Let $x_{n+1}=x_n+2y_n$ and $y_{n+1}=y_n-x_n$, where $x_1=1$ and $y_1=-1$. I tried proving by contradiction, I tried by induction, I got nothing. This is a question I had on an exam, I didn't manage ...
2
votes
1answer
66 views

What's an intuitive way to compute summation of this series?

What's an intuitive way to compute $$\log(1)+\log (2)+\log (3)+\cdots+\log (n-1)+\log (n)$$ or for $n>a$ $$\log(a)+\log (a+1)+\log (a+2)+\cdots+\log (n-1)+\log (n) $$ I know the answer for ...
3
votes
1answer
44 views

investigate $\sum\limits_{n\ge1}{\frac{(-1)^n}{n^\alpha \ln n}}$

I need to investigate the series (Hence, when the series converges and when the series converges absolutely depending on $\alpha$). $$\sum\limits_{n\ge2}{\frac{(-1)^n}{n^\alpha \ln n}}$$ For ...
4
votes
1answer
114 views

I don't understand the solution to this limit.

$\lim _{n\to \infty }\left(\frac{1+5+5^2+...+5^{n-1}}{1-25^n}\right)$ I have a solution to this question, but I don't really understand it. It's: $\lim _{n\to \infty \:}\left(\frac{1+5+5^2+\ldots ...
2
votes
1answer
38 views

$\lim_{n \rightarrow \infty} \frac{3}{n} \sum_{k=1}^n \left(\frac{2n+3k}{n} \right)^2$

How to solve this: $$\lim_{n \rightarrow \infty} \frac{3}{n} \sum_{k=1}^n \left(\frac{2n+3k}{n} \right)^2$$ The answer is supposed to be 39. My attempt: ...
4
votes
1answer
33 views

A question about the convergence of series.

If$\hspace{0.3cm}$ $\sum a_n$ and $\hspace{0.3cm}$$\sum b_n$$\hspace{0.3cm}$ are convergent then which of the following is true? $1.$$\hspace{0.3cm}$$a_{n+1}<a_n$$\hspace{0.3cm}$ $\forall n$ ...
2
votes
1answer
49 views

Series in a space which is not complete

Let $X$ be a normed vector space and $\left\lbrace x_n \right\rbrace_{n \in \mathbb{N}} \in X^{\mathbb{N}}$ with $$\sum_{n=1}^{\infty} \|x_n\| < \infty \wedge \sum_{n=1}^{\infty} x_n \notin X,$$ ...
2
votes
0answers
100 views

Find the function whose Taylor series is $\log(x)+\log(x+1)+\log(x+2)+\ldots$

How do I find a function $f$ whose Taylor series is $$\log(x)+\log(x+1)+\log(x+2)+\ldots$$ for some point $x=a$? It would seem that $$\left.\frac{\partial^n}{\partial x^n}f(x) \right|_{x=a} = ...
2
votes
0answers
43 views

Number of compositions, does this sequence have a closed form?

Does this sequence have a closed form: 1, 1, 2, 3, 5, 8, 14, 25, 46, 87, 167, 324, 634, 1248, 2466, 4887, 9706, 19308, 38455, 76659, 152925, 305232, 609488, 1217429, 2432399, 4860881, 9715511, ...
2
votes
4answers
476 views

Numbers whose digits sum to 7

Let $S$ be the sequence of all positive integers whose decimal digits add to exactly 7, in increasing order: $$S = \langle7, 16, 25, \ldots, 70, 106, 115, 124, \ldots 160, 205, \ldots, 10230010, ...
1
vote
0answers
23 views

Find limit property

Let $f$ be a function on $\mathbb{R}$ satisfy: $|f(x)-f(y)|\leq|x-y|$ $\forall x,y\in\mathbb{R}$. Consider the sequence: $$u_{n+1}=\frac{u_n+f(u_n)}{2},u_0=a$$ Research the limit property of this ...
3
votes
5answers
105 views

Taylor expansion of $\frac{1}{1+x^{2}}$ at $0.$

I am trying to find the Taylor expansion for the function $$f(x) = \frac{1}{1+x^{2}}$$ at $a=0.$ I have looked up the Taylor expansion and concluded that it would be sufficient to show that ...
2
votes
3answers
57 views

Find the 325th term of the series 7,16,25,34…

One of my friend gave me the series 7,16,25,34,43... I figured it out easily that the sum of digits is 7 in each case. How can I find the 325th term of this series? Also is there any trick/formula to ...