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

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1answer
20 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.: ...
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0answers
15 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 ...
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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 ...
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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 ...
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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 ...
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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
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4answers
70 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)} - ...
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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 ...
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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 $B$ ...
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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 ...
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0answers
26 views

Finding the sum [on hold]

How to find the sum of $\sum_{n=1}^{\infty}\frac{(-1)^n}{2^nn}$ ?
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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 ...
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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 ...
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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}})} = ...
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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 ...
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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 ...
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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 : ...
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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 ...
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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 $(b_n)$?
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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??
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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 ...
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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 ...
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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 ...
2
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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 ...
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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 ...
4
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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 ...
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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 ...
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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. ...
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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) ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
26 views

How to calculate variant of geometric series based on sequences of Catalan numbers?

I want to calculate $$\sum_{n=0}^{\infty}\frac{(2n)!}{n!(n+1)!}k^{n+1}C$$ where $0<k<1$ and $C$ and $k$ are some constants, and $0<C$. Is there any possible range of $0<C$ that allows ...
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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 ...
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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: ...
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4answers
47 views

Why is $\sum_{r=1}^{m-1} (2r+1)r=\sum_{r=1}^{m-1} 4\binom{r}{2} + 3\binom{r}{1}$?

How did the summation expression get transformed to combination? From where did the constants $4$ and $3$ come from? $$ \begin{align*} T(m^2-1) &= \sum_{r=1}^{m-1} (2r+1)r\\ &= \ ...
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2answers
35 views

What property of summation is used while solving this problem?

Saw this problem on a website. Can someone explain how the summation is split into summation of summation? What property of summation was used here? $$ T(n) = \sum_{k=1}^n \lfloor \sqrt{k} \rfloor. $$ ...
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5answers
88 views

Analysis: Prove divergence of sequence $(n!)^{\frac2n}$

I am trying to prove that the sequence $$a_n = (n!)^{\frac2n}$$ tends to infinity as $ n \to \infty $. I've tried different methods but I haven't really got anywhere. Any solutions/hints?
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0answers
35 views

Find the finite sequence that minimizes the value of $T_5(P)$

Given a finite sequence $P(a_1,b_1),(a_2,b_2),...,(a_n,b_n)$, define $T_1(P):=a_1+b_1$, $\forall 2\leq k\leq n$, $T_k(P)=b_k+\max\{T_{k-1}(P),a_1+a_2+...+a_k\}$. Let $m=\min\{a,b,c,d\}$. ...
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1answer
31 views

Probability of drawing an element from a countably infinite sequence

Consider a sequence containing $A$ and $B$ where, starting at $n=0$, there are $2^n A$'s followed by $2^{n+1} B \ $'s, so the sequence begins $$A, B, B, A, A, B, B, B, B, A, A, A, A, B, B, B, B, B, ...
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1answer
28 views

$k_{n+1}\le (1+2\varepsilon)k_n$ for $k_n:=\lfloor(1+\varepsilon)^n\rfloor$ and $\varepsilon>0$

Let $$k_n:=\lfloor(1+\varepsilon)^n\rfloor\stackrel{\text{def}}{=}\max\left\{k\in\mathbb{Z}:k\le(1+\varepsilon)^n\right\}\;\;\;\text{for }n\in\mathbb{N}$$ How can we prove $k_{n+1}\le ...
2
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2answers
21 views

Show that $\lim_{n\to\infty}\sum_{k=1}^n\bigl|k\bigl(f\bigl(\frac{1}{k}\bigr)-f\bigl(-\frac{1}{k}\bigr)\bigr)-2f'(0)\bigr|$ exists

Suppose $f\in C^3[-1,1]$, show that $$\lim_{n\to\infty}\sum_{k=1}^n\left|k\left(f\left(\frac{1}{k}\right)-f\left(-\frac{1}{k}\right)\right)-2f'(0)\right|$$ exists. I realized that ...
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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 ...
4
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0answers
69 views

An infinite series of integrals $\int_{0}^{\eta}\cos nt\log\left(\frac{\cos(t/2)+\sqrt{\cos^2(t/2) -\cos^2(\eta/2)}}{\cos(\eta/2)}\right) dt$

I am reading a paper (sorry, no e-copy) with a number of infinite series, in which each term of the series is an integral of a complicated transcendental function like the one in the title. There ...
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2answers
124 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 ...
2
votes
1answer
37 views

Why can't the definition of convergence be alterted to this one?

I am trying to find out of a seqence with the following property is convergent: Let $(r_n)$ be a sequence of real numbers. Suppose there is a number $r\in\mathbb{R}$ such that for any ...
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1answer
36 views

sequence defined by norm

Let $(u_n)_n$ be defined by: $\quad \begin{cases}u_1=1 & \\ \\ u_n=\left( \sum\limits_{k=1}^{n}u_{k}\right)^{\frac{1}{2}} & \end{cases} $ Show that $u_{n}\to +\infty$ and $u_n \simeq ...
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 ...
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1answer
81 views

What's the limit of this sum $ S_n=1^s+\frac{1}{2^s}+3^s+ \frac{1}{4^s}+5^s+\frac{1}{6^s}+\cdots+n^s+\frac{1}{(n+1)^s} $ [on hold]

Let $ S_n=1^s+\frac{1}{2^s}+3^s+ \frac{1}{4^s}+5^s+\frac{1}{6^s}+\cdots+n^s+\frac{1}{(n+1)^s} $ and s be a complex variable . $s=\sigma +it $ where $\sigma ,t \in\mathbb{R} $ , Note :I edit the ...