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

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1
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4answers
67 views

Power series for the rational function $(1+x)^3/(1-x)^3$

Show that $$\dfrac{(1+x)^3}{(1-x)^3} =1 + \displaystyle\sum_{n=1}^{\infty} (4n^2+2)x^n$$ I tried with the partial frationaising the expression that gives me $\dfrac{-6}{(x-1)} - ...
1
vote
1answer
18 views

convergent subsequences etc.

It is well known that: A sequence of real numbers converges $\{a_n\}$ to p, if and only if every subsequence $\{a_{n_k}\}$ converges to p. I am wondering if this similar statement holds.: ...
0
votes
0answers
14 views

Estimating the divergence of a “convex series.”

An exercise from R.C. Buck's Advanced Calculus: Let $f≥0$, $f'≥0$, $f'' \geq 0$ for $1≤x<\infty$. Show that $$0≤ \sum_1^n f(k) - \int_1^n f(x)dx - \frac12f(n) - \frac12f(1)≤\frac14f'(n)$$ for ...
1
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2answers
53 views

Let $\{a_n\}_{n=1}^\infty$ be an infinite sequence. Does there exist an infinite series whose partial sums is $\{a_n\}_{n=1}^\infty$?

Definition: By an Infinite Sequence of real numbers, we shall mean any real valued function whose domain is the set of all positive integers. Definition: By an Infinite Series of real numbers, we ...
1
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3answers
53 views

Induction on nth polynomial proof.

Question: Prove by induction that $ 1+r+r^2+\cdots+r^n = \dfrac {1-r^{n+1}} {1-r} $ where $ r \in \mathbb{R} $ When $n$ is odd, this is really easy as the right side breaks down to $\dfrac ...
-1
votes
0answers
35 views

trouble with sequence and series question [on hold]

$x^2f''(x)+xf'(x)+(x^2-1)f(x)=0$ show that $f(x) = \displaystyle\sum_{n=0}^{\infty} \dfrac{(-1)^n}{n!(n+1)!2^{2n+1}}x^{2n+1}$ Dont know to approach this one
-1
votes
0answers
41 views

Sequences problem from AMM

A cute problem I think, but the (official) solutions are somewhat unnatural. Would be interesting in seeing some alternative approaches. Suppose $x_1,x_2,...$ is a sequence of positive real numbers ...
6
votes
2answers
123 views

Find the values of the positive integers $n$ such that: $\frac{(-\sqrt{3}+2)^{n+1}+(\sqrt{3}+2)^{n+1}}{4n+3}$ is positive integer

My question is as follow: Find the values of the positive integers $n$ such that: $$\frac{(-\sqrt{3}+2)^{n+1}+(\sqrt{3}+2)^{n+1}}{4n+3}$$ is positive integer. I can see that for $n=1$ (among some ...
3
votes
5answers
58 views

Proving the convergence of $\sum_{n\geq 2}\log\left(1-\frac{1}{n^2}\right)$

How to prove that the series $$\sum_{n\geq 2}\log\left(1-\frac{1}{n^2}\right)$$ is convergent? What about finding the sum? My attempt: $$\ln (1-1/n^2)= \ln(n-1) -2\ln n + \ln(n+1)$$ The term in the ...
0
votes
2answers
35 views

Showing that the $\lim s_n\neq\dfrac{2}{3}$ when $s_n=\dfrac{2}{3n}$

I'm trying to verify if I what I did to show that limit does not exist is valid using the negation of the definition: $\exists \epsilon>0, \forall N \in \mathbb{N}$ such that $n>N \implies ...
-1
votes
0answers
38 views

Sequence with Conditions and Possible Answers

Given that $a_1$, $a_2$, $a_3$, . . . $a_n$ is a sequence of positive real numbers such that: For all positive integers $m$ and $n$, $a_{mn}$ = $a_m$$a_n$, AND there exists a positive real number $B$ ...
1
vote
1answer
75 views

use comparison test to show divergence or convergence

I'm not sure if my reasoning is correct. a) $\displaystyle \sum_{n=2}^{\infty} \frac{\ln^5{(2n^7+13)+10\sin(n)}}{n\cdot \ln^6{(n^\frac{7}{8}}+2\sqrt{n}-1)\cdot\ln{\ln{(n+(-1)^n}})} = ...
2
votes
3answers
60 views

Limit of $a_n$, where $a_1=-\frac32$ and $3a_{n+1}=2+a_n^3$.

Let $a_1=-\frac32$, and $3a_{n+1}=2+a_n^3$. I need to show that $\displaystyle \lim_{n\to \infty} a_n = 1$. I can show that the sequence is monotonically increasing and bounded as follows: By the ...
0
votes
4answers
59 views

How to prove that the sum of binomials equals $\begin{pmatrix}2n\\n\end{pmatrix}$ [duplicate]

I've stumbled upon this lemma a few times in my textbook: $$\sum_{k=0}^{n}\begin{pmatrix}n\\k\end{pmatrix}=\begin{pmatrix}2n\\n\end{pmatrix}$$ I've been trying to prove it, but I simply can't seem to ...
-5
votes
0answers
26 views

Finding the sum [on hold]

How to find the sum of $\sum_{n=1}^{\infty}\frac{(-1)^n}{2^nn}$ ?
0
votes
2answers
41 views

examine if series is convergent

I have problem with $$ a_n=\sum_{n=2}^{\infty}(-1)^{\left\lfloor{\frac{n^3+n+1}{3n^2-1}} \right\rfloor}\cdot\frac{\ln(n)}{n}$$ I'd like to use here a dirichlet's test I know how to show ...
1
vote
0answers
25 views

Switch integral and sum for Bessel function.

I haven't real knowledge in Bessel's function and I'd like to know how to switch integral and sum in these two equations. I've already tried a lot of ideas but nothing really works. The first one is : ...
5
votes
1answer
133 views
+50

How find this sequence recursive relations

Question: Let $$A_{n}=\sum_{i=0}^{n-3}(-1)^{n+i-2}\dfrac{13n^2-31n-10ni+9i+i^2+16}{(3n-i-3)(3n-i-4)(2n-i-3)!\cdot i!}$$ I want find the $A_{n}$ recursive relations,such as following form ...
1
vote
1answer
32 views

Show that the series $∑_{m=1}^{∞}(r^{-m}/(2^{m}-1))$ is convergente for some positive integer $r>0$

The Erdős-Borwein Constant can be found in http://mathworld.wolfram.com/Erdos-BorweinConstant.html My question is : Show that the series $$∑_{m=1}^{∞}r^{-m}/(2^{m}-1)$$ is convergente for some ...
8
votes
2answers
229 views
+50

Showing that $\sum_{k=0}^{\infty} a^{k} \cos(kx) = \frac{1- a \cos x}{1-2a \cos x + a^{2}}$ without using complex variables

I was challenged a couple of weeks ago by Ron Gordon to show that $$\sum_{k=0}^{\infty} a^{k} \cos(kx) = \frac{1- a \cos x}{1-2a \cos x + a^{2}} \ \ (|a| <1)$$ without using complex variables. ...
4
votes
0answers
27 views

How to get the nine cycles without trial and error?

Determine the nine cycles that occur in sequences of natural numbers where each succeeding term is the sum of the cubes of the digits of the previous number. My approach is to try one-by-one starting ...
0
votes
3answers
37 views

fractional part of the square of natural number

How can if prove that the sequence :$$a_n\:=\left\{\sqrt{n}\right\}\left(fractional\:part\:of\:\sqrt{n}\right)\:=\:\:\sqrt{n}\:-\:\left[\sqrt{n}\right]$$ is bounded from above by 1? So far i try ...
6
votes
2answers
98 views

Evaluating sums using residues $(-1)^n/n^2$ [duplicate]

I am an alien towards compelx analysis, with very little know I am posing a question, who someone may want to help with. Evaluate: $$\frac{1}{4}\cdot \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$$ In ...
0
votes
1answer
58 views

Convergence and limit of Muller sequence

The Muller sequence is given by the recursive definition: $U(n+1)=111-\frac{1130}{U(n)}+\frac{3000}{U(n)U(n-1)}$ with $U(0)=5.5$ and $U(1)=\frac{61}{11}$. This sequence is interesting in ...
2
votes
0answers
19 views

How to prove there exists a positive integer $k,d$ such $\sum_{i=1}^{2k}a_{id}=k-2014$

Let $\{a_{n}\}_{n=1}^{\infty}$ be a non-negative integer such that: for any postive integer $m,n$, we have: $$\sum_{i=1}^{2m}a_{in}\le m.$$ Show that there exist positive integers $k,d$ such that: ...
0
votes
0answers
20 views

The condition of uniform convergence of $\sum a_n\sin(nx)$ [on hold]

If $a_n$ satisfy: $a_n \geq a_{n+1}$, and $a_n \rightarrow 0$ as $n \rightarrow +\infty$, show that: $$\sum_{n=1}^{\infty}a_n\sin(nx)$$ is uniform convergence in $\Bbb{R}$ if and only if $$\lim_{n ...
3
votes
1answer
63 views

In $\triangle ABC$ , find the value of $\cos A+\cos B$

The sides of $\triangle$ABC are in Arithmetic Progression (order being $a$, $b$, $c$) and satisfy $\dfrac{2!}{1!9!}+\dfrac{2!}{3!7!}+\dfrac{1}{5!5!}=\dfrac{8^a}{(2b)!}$, Then prove that the value of ...
1
vote
4answers
134 views

Absolute convergence in a metric space

Let $(X,d)$ be a metric space, $(a_n)$ and $(b_n)$ are sequences in $(X,d)$. If $\sum_{n=1}^\infty d(a_n,b_n)$ is absolutely convergent, what do I say about the convergence of $(a_n)$ and $(b_n)$?
5
votes
2answers
323 views

Convergence of a sequence of real numbers

Let $\alpha, \gamma$ be real numbers such that $0<\alpha<1$ and $\gamma>0$. Consider the sequence of real numbers given by $$ \begin{cases} x_0\ne 0&\\ ...
2
votes
2answers
98 views

Find $a_{2012}-3a_{2010}/3 a_{2011}$ where the sequence $a_n$ is determined by roots of a quadratic equation

If $\alpha$ and $\beta$ are the roots of $x^2-9x-3=0$, $a_n=\alpha^n-\beta^n$ and $b_n=\alpha^n+\beta^n$, then find the value of $\dfrac{a_{2012}-3a_{2010}}{3 a_{2011}}$ and ...
4
votes
1answer
85 views

Show that series converge or diverge

If $\displaystyle \sum_{n=1}^{\infty} a_n$ converge and has positive terms then decide if following series converge or diverge : a) $\displaystyle \sum_{n=1}^{\infty} a_n \cdot \sin{a_n}$ I think it ...
0
votes
2answers
27 views

Find a recursive definition for the sequences

The first sequence given is 3, 7, 16, 41, 77,.... I really am quite stuck on this because I can't seem to find any relationship between one term and the terms prior to it. I first noticed that it ...
21
votes
3answers
2k views

Definition of convergence of a nested radical $\sqrt{a_1 + \sqrt{a_2 + \sqrt{a_3 + \sqrt{a_4+\cdots}}}}$?

In my answer to the recent question Nested Square Roots, @GEdgar correctly raised the issue that the proof is incomplete unless I show that the intermediate expressions do converge to a (finite) ...
0
votes
1answer
179 views

Simple interest equated monthly installments while borrowing then paying back

What is the formula for calculating equated monthly installments with simple interest? How can one derive it?
0
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0answers
19 views

Merely conditionally convergent series [on hold]

What is the definition of 'Merely conditionally convergent series'? Is it exactly same as 'conditionally convergent series'? Or different??
13
votes
1answer
331 views

Another Ramanujan's formula dealing with $\coth^{2}(5\pi)$

Ramanujan mentions in one of his letters to Hardy that $$\frac{1^{5}}{e^{2\pi} - 1}\cdot\frac{1}{2500 + 1^{4}} + \frac{2^{5}}{e^{4\pi} - 1}\cdot\frac{1}{2500 + 2^{4}} + \cdots = ...
2
votes
4answers
325 views

Evaluate $ \lim\limits_{x\to 1} \frac{(\ln x)^2}{1-x} $

I have trouble finding the limit of the following : $$ \lim\limits_{x\to 1} \frac{(\ln x)^2}{1-x} $$ using the rule from L´Hopital. Since both quotients converge to $0$, I should be able to use ...
0
votes
2answers
398 views

Question on compound interest

If you deposit $100 at the end of every month into an account that pays 3% interest per year compounded monthly,the amount of interest accumulated after months is given by the sequence. I tried the ...
2
votes
2answers
66 views

Stone-Weierstrass: Examples

By Stone-Weierstrass one has: $f\in\mathcal{C}(K):\quad p_n\to f$ Now, for analytic functions this is just Taylor: $$f\in\mathcal{C}^\omega([a,b]):\quad f(x)=\sum_{k=0}^\infty a_kx^k$$ But, how does ...
0
votes
2answers
34 views

Absolute convergence of an infinite series and p-series test

Why does the infinite series $\sum_{n=1}^{\infty}\frac{(-1)^n}{\sqrt n}z^n$ where $z\in \mathbb{C}$ converge absolutely for $|z|<1$. Doesn't the series diverge because if we apply the absolute ...
0
votes
0answers
23 views

Domain of convergence of series

Could you help me to find the domain of convergence of series : $$\sum\limits_{n,m=1}\frac{n}{m!}z_1^nz_2^m$$ in $\mathbb{C}^2$. The series is product of two series. I think the answer is ...
22
votes
1answer
299 views

Is this integral $\int_0^1\left(\left\{\frac1x\right\}-\frac12\right)\frac{\log(x)}{x}dx$ equal to zero?

My initial question was to find if this integral $$ \int_0^1 \left(\left\{\frac 1x\right\}-\frac12\right)\frac{\log(x)}{x}dx$$ is convergent or divergent. ($\left\{\frac 1x\right\}$ is the fractional ...
3
votes
3answers
179 views

Proving convergence of a series. Is my proof correct?

Prove that if $\sum_{n=0}^{\infty}{a_{2n}}$ and $\sum_{n=0}^{\infty}{a_{2n+1}}$ are convergent series then $\sum_{n=0}^{\infty}{a_{n}}$ is also convergent From the assumption we know that ...
-1
votes
1answer
62 views

Decide convergence of the series .

I have problem with these two: a) $\displaystyle \sum_{n=2}^{\infty} \frac{1}{(\ln{\ln{n}})^{\ln{n}}}$ b) $\displaystyle \sum_{n=3}^{\infty} \frac{1}{n \cdot \ln{n} \cdot \ln{\ln{n}}}$ My try: a) ...
1
vote
3answers
68 views

Why does $1+p+p^2+\dotsb+p^{n-1}=\frac{1-p^n}{1-p}$ [duplicate]

$$y_n=\rho^ny_0+(1+\rho+\rho^2+\cdots+\rho^{n-1})b.$$ If $\rho \not=1$, we can write this solution in the more compact form $$y_n=\rho^ny_0+\frac{1-\rho^n}{1-\rho}b.$$ This is from Elem. Diff. ...
1
vote
1answer
25 views

Find all parameters for for which the series is convergent - checking

I'm not sure if my reasoning is good. Find all parameters $a$ for for which the series is convergent $\displaystyle \sum_{n=1}^{\infty}a^{w_n}$ where $\displaystyle w_n=(\sqrt[n]{2}-1)^{(-1)}$ My ...
-1
votes
4answers
41 views

Why does $\lim_{n \to \infty} \sum_{k=1}^n\frac{t^{k+1}}{(k+1)!}=e^t-t-1$?

Why does $\lim_{n \to \infty} \sum_{k=1}^n\frac{t^{k+1}}{(k+1)!}=e^t-t-1$? I know $\lim_{n \to \infty} \sum_{k=0}^n\frac{t^k}{k!}=e^t$, but my sum starts at $k=1$ and also has ...
1
vote
2answers
33 views

Find an arithmetic sequence which…

Find an arithmetic sequence with $5$ terms which sum of them are $15$ and if multiply all terms the answer would be $1155$ $a$ is the first term. So $a(a + d)(a + 2d)(a + 3d)(a + 4d) =1155$ And ...
0
votes
1answer
32 views

Find the general formula of this sequence

Here is the sequence 1 2 2 4 4 6 6 10 10 14 14 20 20 26 26 36 36 46 46 60 60 ··· I just know the recursion formula:$\displaystyle ...
6
votes
0answers
42 views
+50

Product of permutations of consecutive numbers yields arithmetic sequence

Let $n\geq 3$ be an integer, and $a,b$ be positive integers. Let $c_1,\ldots,c_n$ be a permutation of $a,a+1,\ldots,a+(n-1)$, and $d_1,\ldots,d_n$ be a permutation of $b,b+1,\ldots,b+(n-1)$. Is it ...