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

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4answers
21 views

Function as a series :

Let $f(x)=\sum_{n=0}^{+\infty}\dfrac{x^n}{n!}$. Verify that $$\int_0^xf(x)dt=f(x)-1$$ This is the exercise 3 of the section $7.4$, of Guidorizzi's Calculus, Vol. 4. What I have tried: By the ratio ...
2
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1answer
25 views

Proving Archimedes Sequences Equal $\pi$.

I encountered the following problem in my text An Introduction to Analysis by William Wade. It was the last problem in the section and has an * next to it. I'm not sure if this indicates a challenge ...
0
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0answers
15 views

Condition for derivative sequence to converge?

Let $E_n(R)$ be a sequence of function that converge to $E(R)$, When we can said that $\dot{E}_n(R)$ converge to $\dot{E}(R)$. I can assume: uniform converge of $E_n$ to $E$. $E(R)$ convex. ...
0
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1answer
23 views

How to know if a space has a convergent subsequence?

So this is something I have been struggling with lately... how do we generally know that a space/set has a subsequence that converges? The current one I am struggling with is the space of sequences ...
0
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1answer
30 views

Proving a sequence is increasing and converging as $n\to \infty$.

Suppose that $x_0 \in (-1,0)$ and $x_n=\sqrt{x_{n-1}+1}-1$ for $n \in \mathbb N$. Prove that $x_n \uparrow 0$ as $n\to \infty$. What happens when $x_0 \in [-1,0]$? Before this, the problems I did had ...
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2answers
53 views

Closed form of $\sum_{n=1}^{\infty} \frac{J_0(2n)}{n^2}$

I'm new in the area of the series involving Bessel function of the first kind. What are the usual tools you would recommend me for computing such a series? Thanks. $$\sum_{n=1}^{\infty} ...
0
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1answer
19 views

Geometric series for values between 0 and 1

I am given that geometric series is defined as the following $1-x+x^2-x^3+x^4$ for values in range $0<x<1$. I am also told expected value can be calculated by using the following equation: ...
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2answers
18 views

Convergence of a sequence of functions involving rational and irrational numbers

For each $n\in \mathbb N$, let $f_n(x)=lim_{m\rightarrow \infty} (cosn!\pi x)^{2m}$ Show that the sequence converges on $\mathbb R$ to the function $f$ defined by $$f(x)=1, x\in \mathbb Q$$ $$=0, ...
0
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3answers
52 views

Does this sequence diverge to ∞?

The sequence $(a_n)_{n \geq 1}$ is defined as follows: $$a_n:= \begin{cases} 0 \quad \text{if} \quad n \quad \text{is odd}\\ n \quad \text{if} \quad n \quad \text{is even}\end{cases} \quad .$$ Does ...
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3answers
95 views

Limit of the sequence $\frac{1^k+2^k+…+n^k}{n^{k+1}}$ [duplicate]

How would someone find the limit of the sequence $a_n = \frac{1^k+2^k+...+n^k}{n^{k+1}}, k \in \mathbb{N}$ as $n$ goes to Infinity? Can someone give me maybe a hint where to start?
5
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2answers
75 views

Computing in closed form $\sum_{n=1}^{\infty}\frac{\operatorname{Ci}\left(\frac{3}{4}\zeta(2) \space n\right)}{n^2}$

What tools would you recommend me for computing the series below? $$\sum_{n=1}^{\infty}\frac{\operatorname{\displaystyle Ci\left(\frac{3}{4}\zeta(2) \space n\right)}}{n^2}$$ I lack the starting ...
5
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3answers
32 views

Convergence of a series with general term $u_n=\int_0^{\infty}e^{-x^n}dx$

I would like to find if the series $\displaystyle \sum_{n=1}^{\infty}u_n$ is convergent or divergent where $$u_n=\int_0^{\infty}e^{-x^n}dx. $$ I've tried to obtain $v_n$ with explicit form such that ...
2
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1answer
48 views

How to find this double summation?

To find the value of $$\sum_{m=1}^{∞}{\sum_{n=1}^{∞}{\frac{m^2\cdot n}{3^m \cdot (n\cdot 3^m+m\cdot3^n)} } }$$ I dont know how to proceed to these kind of problems. Can anybody provide a sol to this ...
1
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5answers
105 views

Does $\sum_{n=1}^\infty \frac{\cos(\ln(n))}{n}$ converge? [duplicate]

EDIT: the question is answered here Divergence of $\sum\limits_{n=1}^{\infty} \frac{\cos(\log(n))}{n}$ Using integrals, I managed to prove that $$\displaystyle \forall m, \sin(\ln(m+1))\leq ...
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0answers
19 views

Prove that this sequence of fractions is returned unchanged after the divisor recurrence, the matrix inverse, and the sum over divisors.

Consider the fraction of binomial coefficients and powers of $4$: $$a(n)=\frac{\binom{2 (n-1)}{n-1}}{4^{n-1}}$$ starting: ...
1
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1answer
154 views

True or False. Convergent subsequence

Is the statement true or false? If $(x_n)$ has a convergent subsequence,then $(x_n)$ is bounded. The statement is False. However, can someone please show me an example of a sequence with ...
0
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0answers
29 views

Condition for a function $f: \mathbb R \rightarrow \mathbb R$ being right or left-continuous at $a \in \mathbb R$.

I know that $f: X \rightarrow \mathbb C$ is continuous if and only if for every convergent sequence $(x_n)$ in $X$ the identity holds $\lim_{n \rightarrow \infty} f(x_n) = f(\lim_{n \rightarrow ...
0
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1answer
24 views

Infinite summation of exponential $\sum_{n\in\mathbb{N}}e^{-n^k}$

For interger $k\geq 2$ is it possible to compute the sum and get an expression in terms of $k$? $\sum_{n\in\mathbb{N}}e^{-n^k}$
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2answers
105 views

Sum of the series $\frac{1}{2\cdot 4}+\frac{1\cdot3}{2\cdot4\cdot6}+\dots$

I am recently struck upon this question that asks to find the sum until infinite terms $$\frac{1}{2\cdot 4}+\frac{1\cdot3}{2\cdot4\cdot6}+\frac{1\cdot3\cdot5}{2\cdot4\cdot6\cdot8}+.....∞$$ I tried my ...
0
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3answers
32 views

sequence consisting of finite number of distinct values.

This is a question from my previous year analysis exam .The question says : Can we construct a sequence which converges but never attains its limits,such that its terms consist of a finite number ...
0
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2answers
15 views

Sequence converging to one.

Suppose that x_0 is a real number and x_n = [1+x_(n-1)]/2 for all natural n. Use the Monotone Convergence Theorem to prove x_n → 1 as n grows. Can someone please help me? I don't know what to assume ...
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3answers
82 views

Sum of the infinite series $\frac16+\frac{5}{6\cdot 12} + \frac{5\cdot8}{6\cdot12\cdot18} + \dots$

We can find the sum of infinite geometric series but I am stuck on this problem. Find the sum of the following infinite series: $$\frac16+\frac{5}{6\cdot 12} + \frac{5\cdot8}{6\cdot12\cdot18} + ...
0
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1answer
24 views

A geometric series weighted by a telescoping series

Is is possible to get a closed form expression of the sum of the series shown below: $$S_n = 1 + (a_1 - a_0)r + (a_2- a_1)r^2 + ... + (a_n - a_{n-1})r^n$$ where, $0 < a_0 < a_1 < ... < ...
2
votes
1answer
28 views

Sequence, Monotone Convergence Theorem.

Suppose that $x_0 \geq 2$ and $x_n = 2 + \sqrt{x_{n-1} - 2}$ for all natural $n$. Use the Monotone Convergence Theorem to prove that either $x_n \rightarrow 2$ or $x_n \rightarrow 3$ as $n$ grows. ...
1
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1answer
18 views

Decreasing sequence, bounded below.

Suppose that $0 \leq x_1 < 1$ and $x_{n+1} = 1 - \sqrt{1 - x_n}$ for all natural $n$. Prove that $x_n$ is decreasing and bounded below as $n$ converges. Attempt: Suppose that $0 \leq x_1 < 1$ ...
1
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1answer
27 views

Does the sequence converge or diverge?

I'm having trouble understanding how to do this one. If anyone could help I would be grateful. Does the sequence $$ \left\{ \sum_{n=1}^k \left( \frac{1}{\sqrt{k^2+n}} \right) ...
0
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2answers
24 views

Understanding and writing limit proofs

I got this question : Consider $a_n,\:b_{n\:}$ sequences such that for every n , $0\le a_n\le \:b_{n\:}$. Let $\lim _{n\to \infty }\left(\frac{b_n}{a_n}\right)\:=\:1$, and $a_n$ is a ...
0
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1answer
21 views

Prove boundedness of recurrence relation

For a number sequence $\{y_n\}$ we know that $y_{n+1} = 2y_n-y^2_n$ If: $0<y_0<1$ show that $0<y_n<1$ for all integers $n>0$ I've tried solving the recurrence relation, but I couldn't ...
1
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1answer
57 views

Why do both trig functions have the same Macluarin series?

Both the degree version and the radian version of the trig functions have the same Maclaurin series, yet they are different. How is this possible? How can two different functions have the same ...
1
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0answers
17 views

Sequence terms being divisible

Here's a question I would like hints for: The sequence ${x_n}$ is defined by $x_{n+2}=6x_{n+1}-9x_{n}$ for $n \geq 0$ where $x_0=3$ and $x_1=18$. What is the smallest $k$ such that $x_k$ is ...
1
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1answer
14 views

Series expansions and perturbation

My professor said that $ f \left( y_1(x)+ \epsilon y_2(x)+... \right)= f(y_1(x)) +f'(y_1(x))\> (\epsilon y_2(x)+...) + ...$ but I have no idea how the series continues. Has anyone seen this ...
3
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2answers
107 views

Proving the bound $\left ( 1+\frac{x}{n} \right)^n \leqslant 3^x$, $\forall x \in \mathbb{R^+}$

I'm trying to directly prove the above bound. I have tried expanding it $$\left ( 1+\frac{x}{n} \right)^n = \sum_{k\geqslant 0} \binom{n}{k}\left ( \frac{x}{n}\right)^k$$ $$= \sum_{k=0\dots n ...
0
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0answers
78 views

Is it possible to determine the value of the following function?

Let $a_n$ and $b_n$ be a pair of generic sequences, and let $L$ be a constant. Consider the following function: $$f(x):=a_o+\sum_{n=1}^{\infty}\left({a_n}\cos{\frac{n\pi x}{L}+b_n\sin{\frac{n\pi ...
0
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1answer
35 views

Geometrical Sequence

I have a sequence $U(n)$ with $U(n+1) = f(U(n))$ with $f(x) = ax+b$. I'm supposed to express $U(n)$ as a function dependant on $n$. Doing so with an auxiliary sequence $V(n) = U(n) - ɑ$. Where $ɑ$ ...
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1answer
27 views

What sequences could satisfy these requirements?

I need to find a sequence which converges to $0$ but is not in any space $\ell^p$, where $1 \leq p < +\infty$. And, I need to find a sequence which is in every space $\ell^p$ with $p > 1$ but ...
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3answers
36 views

Common ratio of a GP

Question: If $f$ is a function satisfying $f(x + y) = f(x)f(y)$ for all $x,y$ that are natural numbers; such that $f(1) = 3$ and $$\sum^{n}_{x=1}f(x) = 120$$ find the value of n. I don't understand ...
0
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0answers
47 views

Convergence of the series $\sum_{n=1}^\infty (1+\frac{1}{\sqrt{n}})^{-n^\frac{3}{2}}$

Test the convergency of the series $$\sum_{n=1}^\infty \left(1+\frac{1}{\sqrt{n}}\right)^{-n^\frac{3}{2}}.$$ We know that, if $\sum_{n=1}^\infty U_n$ is convergent, then ...
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4answers
525 views

Baffled by resolving number list

My son's Maths homework was to do with number patterns/sequences. "What is the nth term?". He'd done very well, but the last sequence was something like this: ...
6
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1answer
61 views

Closed-form of $\displaystyle\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\Psi_3(n+1)$

Does the following series have a closed-form \begin{equation} \sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\Psi_3(n+1) \end{equation} where $\Psi_3(x)$ is the polygamma function of order 3. Here is ...
0
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0answers
10 views

If $\lim_{n \to ∞} z_n=L_1$ then $\lim_{n \to ∞} \overline{z_n}=\overline{L_1}$ for $z_n \in \mathbb{C}$

I tried proving that if $\lim_{n \to ∞} z_n=L_1$ then $\lim_{n \to ∞} \overline{z_n}=\overline{L_1}$ for $z_n \in \mathbb{C}$. This is my attemt. Let $\epsilon>0$. Then there exists $N\in ...
2
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3answers
53 views

Sum of Harmonic Numbers

Similar to this question , let $H_n$ be the $n^{th}$ harmonic number, $$ H_n = \sum_{i=1}^{n} \frac{1}{i}$$ Is there a similar method to calculate the following?: $$\sum_{i=1}^{n}iH_i$$
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0answers
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+50

Cauchy product $\sum_{n=-\infty}^{\infty}\sum_{k=-\infty}^{\infty}a_{n-k}b_k$

I have been told that, if $\{a_n\}_{n\in\mathbb{N}}$, $\{a_{-n}\}_{n\in\mathbb{N}^+}$, $\{b_n\}_{n\in\mathbb{N}}$ and $\{b_{-n}\}_{n\in\mathbb{N}^+}$ are absolutely summable complex sequences, ...
3
votes
1answer
66 views

Show that $\lim_{n \to \infty}\prod_{k = n}^{2n}\dfrac{\pi}{2\tan^{-1}k} = 4^{1/\pi}$

MathWorld states that (see equation $(130)$) $$\lim_{n \to \infty}\prod_{k = n}^{2n}\dfrac{\pi}{2\tan^{-1}k} = 4^{1/\pi}$$ and attributes it to Gosper. I believe an approach to establish the formula ...
1
vote
1answer
29 views

Convergent Series $\frac{1}{n^q}, \ \ q>1$

How to show the following result about series? Thank you! Convergent Series: $$\sum_n \frac{1}{n^q}, \ \ q>1$$
0
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3answers
20 views

Bounded Sequences and Extrapolation of Convergence From Related Sequences

I'm considering some sequence $S_n$ which is bounded, and I want to prove that $S_n/n$ is convergent. I'm thinking that I could simply take $lim_{n \to \infty} S_n/n$ and simplify this to $(lim_{n \to ...
1
vote
1answer
31 views

Increasing sequence and converging to zero.

Suppose that $x_{0}$ is between $(-1,0)$ and $$x_{n} = \sqrt{x_{n+1} + 1} - 1$$ for all natural $n$. Prove that $x_{n}$ is increasing and converging to zero as $n$ grows. Can someone please help me ...
0
votes
0answers
27 views

Problems with convergence in mean

I hope you can help me with the following problem Let $\{ e_i : i\in \mathbb{Z}\}$ be and independent U.I. sequence of scalar random variables with zero mean. Let $\{ A_j : j \geq 0\}$ be a sequence ...
0
votes
0answers
17 views

show the AR process is non-stationary. [on hold]

A first order auto-regressive (AR) process $u(n)$ satisfies the equation: $$ u(n) + a_1 u(n-1) = v(n) $$ where $a$ is a constant and $v(n)$ is a white-noise process with variance $\sigma^2_v$. Show ...
0
votes
1answer
23 views

Alternate expression for finite summation

"How many arithmetic operations are required to directly compute $$y=1+x+x^2+...+x^{1023}$$ Use a formula for the sum to come up with an alternate expression for $y$, and show that only 10 ...
3
votes
2answers
40 views

Limit of a sequence proof by contradiction

Suppose I have a monotonically decreasing sequence $a_{n}$ such that $a_{n}$ is positive for all $n \in \mathbb{N}^{+}$ and that $$\lim_{n\rightarrow \infty} \frac{a_{n+1}}{a_{n}} = 0.$$ It seems to ...