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

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56 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
18 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
42 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
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0answers
41 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
28 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
60 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
22 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 ...
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1answer
70 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
176 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
66 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
60 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
36 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
24 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 ...
2
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4answers
320 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 ...
1
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1answer
24 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 ...
1
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2answers
32 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
13 views

How prove exsit postive integer $k,d$ such $\sum_{i=1}^{2k}a_{id}=k-2014$

let $\{a_{n}\}_{n=1}^{\infty}$ is non-negative integer,and such: for any postive integer $m,n$ have $$\sum_{i=1}^{2m}a_{in}\le m$$ show that: there exsit postive integer $k,d$ such ...
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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? $$ T(n) = \sum_{k=1}^n \lfloor \sqrt{k} \rfloor. $$ ...
2
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5answers
86 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
34 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
30 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, ...
2
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1answer
25 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 ...
1
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1answer
25 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
68 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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1answer
102 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
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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
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1answer
61 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
80 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 ...
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1answer
29 views

Limit sup and inf hint

I have problem in finding the Limsup and liminf for the following sequences. Any hint pls? $(s_n) = [1-r^n]\sin \frac{n\pi}{2}$ and $(s_n) = [(-1)^n + 1]n^2$.
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1answer
48 views

Evaluate this infinite product involving $a_k$

Let $a_0 = 5/2$ and $a_k = a_{k-1}^{2} - 2$ for $k \ge 1$ Compute: $$\prod_{k=0}^{\infty} 1 - \frac{1}{a_k}$$ Off the bat, we can seperate $a_0$ $$= -3/2 \cdot \prod_{k=1}^{\infty} 1 - ...
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1answer
47 views

Elementary proof $\sum_{n> N}\frac{1}{n^2} < 1/N $

Is there an elementary proof (not using integrals) $$\sum_{n> N}\frac{1}{n^2} < 1/N $$ for this sum?
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0answers
50 views

Seperating single integral into an double integral.

Please refer to : How to prove that $\int_{0}^{\infty}\sin{x}\arctan{\frac{1}{x}}\,\mathrm dx=\frac{\pi }{2} \big(\frac{e-1}e\big)$ The answer by @Venus. What is the procedure in converting that ...
1
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4answers
100 views

How to prove $ \lim_{n\to\infty}\sum_{k=1}^{n}\frac{n+k}{n^2+k}=\frac{3}{2}$?

How to prove that $\displaystyle \lim_{n\longrightarrow\infty}\sum_{k=1}^{n}\frac{n+k}{n^2+k}=\frac{3}{2}$? I suppose some bounds are nedded, but the ones I have found are not sharp enough (changing ...
2
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2answers
64 views

Computing the limit of an alternating series,

I am looking at the series $$ \sum_{n=1}^\infty\frac{(-1)^n}{n}.$$ This series converges (conditionally) by the alternating series test. How can I compute its limit, which is equal to -log(2)? a) ...
2
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1answer
43 views

On a $\epsilon$-$n$ proof of a limit of a sequence of functions.

Put $\delta_n = 2^{-n}$. To each positive integer $n$ and each real number $t$ corresponds a unique integer $k = k_n(t)$ that satisfies $k\delta_n \le t< (k+1) \delta_n$. Define $$ \psi_n(t) = ...
4
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1answer
89 views

Adding Two Power Series if their bounds are different

I have the following product $$\frac{1}{6}\sum_{n=0}^{\infty}\left(\frac{-x}{2}\right)^n\sum_{n=0}^{\infty}\frac{a_nx^{n+2}}{n!}$$ Where $a_n$ is an arbitrary coefficient. If I factor out $x^2$ ...
4
votes
2answers
73 views

Finding $\lim_{x\rightarrow 0.5^{-}}\sum_{n=0}^{\infty }(-1)^n\left(\frac{x}{1-x}\right)^n$

Finding $$\lim_{x\rightarrow 0.5^{-}}\sum_{n=0}^{\infty }(-1)^n\left(\frac{x}{1-x}\right)^n$$ When I want to use the geometric series, I had a problem with $(-1)^n$ so I stoped.
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0answers
36 views

Clarifying a step in an integration solution

In the accepted answer here, the first two steps in computing the integral are \begin{align} \mathcal{I} =&\frac{1}{2}\int^\infty_0\ln(1-e^{-2x})\ln\left(\frac{x^2}{\pi^2+x^2}\right)\ {\rm d}x\\ ...
2
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1answer
43 views

Determining a radius convergence of a power series

Let $$ \sum_{n=0}^\infty \frac{(-1)^n}{3n+1} x^{3n+1} $$ Is there an immediate way to determine $R=1$?
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0answers
26 views

Show existence of a sub-sequence $(f_{n_k})$ which is uniformly convergent to a function in $C[0,1]$

Let $f_n:[0,1]\rightarrow R$ be a sequence of continuously differentiable function, Let $M>0$ be such that for any $0\le x \le 1$ and natural $n$, $|f_n(x)|$, $|f'_n(x)|<M$ Show existence of a ...
2
votes
1answer
50 views

sum of infinte series with exponential and factorial terms

Want to sum the following series: $$ \sum_{t=1}^\infty e^{-tk} \frac{(tk)^t}{t!} $$ where $k$ is an integer $>0$.
1
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1answer
37 views

Series with $n$th term having integer raised to the power of $n$ in the denominator

$$ 1+\frac{4}{6}+\frac{4\cdot5}{6\cdot9}+\frac{4\cdot5\cdot6}{6\cdot9\cdot12}+\cdots $$ I could reduce it to $n$th term being $\dfrac{(n+1)\cdot(n+2)}{n!\cdot3^n}$. Took me an hour just to get ...
6
votes
3answers
80 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 ...
1
vote
1answer
45 views

An unusual two dimensional sum

Can anyone prove or reference a proof for the following bound (unless it's not true!) $$\sum_{|\underline{k}|_{\infty} > M} \frac{1}{((k_1)^2 + (k_2)^2 )^2} \leq \frac{C}{M^2}$$ where ...
6
votes
4answers
150 views

How to compute $\sum_{n\ =\ 1}^{\infty}\arctan\left(\,\frac{3n^{2}}{ 2n^{4} - 1}\,\right)$

I find this problem on facebook group. $$\mbox{Is it possible to find exact value of}\quad \sum_{n\ =\ 1}^{\infty}\arctan\left(\,\frac{3n^{2}}{ 2n^{4} - 1}\,\right)\ {\large ?}. $$ I think this is ...
1
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1answer
38 views

Absolutely convergent but not convergent

Here, Lemma $2.1$ states that A normed space $X$ is complete if and only if every absolutely convergent series is convergent. I would like to know a series which is absolutely convergent but ...