Recurrence relations, convergence tests, identifying sequences

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0
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26 views

Limit of a recursive sequence with u_n

It is given that $u_{n+1} =1+\frac{1}{u_n}$ and $u_1 =1$. Find the limit of $u_n$ as $n\to\infty$. The limit is $\frac{\sqrt{5}+1}{2}$ from a calculator. Is there an algebraic way to determine ...
0
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2answers
45 views

Sequence $(a_n)$ s.t $\sum\sqrt{a_na_{n+1}}<\infty$ but $\sum a_n=\infty$

I am looking for a positive sequence $(a_n)_{n=1}^{\infty}$ such that $\sum_{n=1}^{\infty}\sqrt{a_na_{n+1}}<\infty$ but $\sum_{n=1}^{\infty} a_n>\infty$. Thank you very much.
2
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3answers
62 views

If $\sum a_n $ is a positive series that diverges, does $\sum \frac{a_n}{1+a_n}$ diverge? [duplicate]

Let $ \{a_n\}_{n=1}^\infty $ be a sequence such that $\displaystyle \sum a_n $ that is divergent to $+\infty$. What can be said about the convergence of $\displaystyle \sum \frac{a_n}{1 + a_n} $? ...
1
vote
1answer
36 views

alternating series test for $\sum_{n=1}^{\infty}(-1)^n\frac{\sqrt{n}}{n+4}$

I must determine if this series converges (using specifically the alternating series test) $$\sum_{n=4}^{\infty}(-1)^n\frac{\sqrt{n}}{n+4}.$$ I know the necessary and sufficient conditions are: The ...
1
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3answers
38 views

On convergence of problematic series.

Determine if the following series is converges or not $$\sum_{n=2}^{\infty}\frac{1}{\ln^{\ln(\ln n)} (n)}$$
2
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1answer
36 views
5
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2answers
44 views

“Nearly” Harmonic Series

It's well known that $$ \sum_{n=1}^{\infty} \frac{1}{n^{1+\varepsilon}} < \infty, \ \forall \varepsilon >0. $$ What happens if we replace $\varepsilon$ with $\varepsilon_n \downarrow 0$? ...
5
votes
2answers
60 views

If $\mathbb f$ is analytic and bounded on the unit disc with zeros $a_n$ then $\sum_{n=1}^\infty \left(1-\lvert a_n\rvert\right) \lt \infty$

I'm going over old exam problems and I got stuck on this one. Suppose that $\mathbb{f}\colon \mathbb{D} \to \mathbb{C}$ is analytic and bounded. Let $\{a_n\}_{n=1}^\infty$ be the non-zero zeros of ...
2
votes
1answer
33 views

Convergence of sequence

Does the following: $$ \begin{align} x_0 & = a \\ x_1 & = x_0 + \frac{1}{1}(x_0 + x_0(c - 1)) \\ x_2 & = x_1 + \frac{1}{2}(b + x_1(c - 1)) \\ x_3 & = x_2 + \frac{1}{3}(b + x_2(c - 1)) ...
1
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3answers
53 views

Showing that $\sum_{n=3}^\infty\frac1{n(\log n)(\log\log n)}$ diverges

I know that the series diverge, I'm just having hard time showing it. $$\sum_{n=3}^\infty\frac1{n(\log n)(\log\log n)}$$ Thanks in advance
3
votes
2answers
35 views

Is a sequence of all the same numbers monotonic?

I'm wondering based on the definition of monotonicity: A sequence where $a_n\geq a_{n+1}$ for all $n\in\mathbb{N}$ is monotonic. So given that the sequence $a_n = 3$ is all the same numbers and ...
2
votes
3answers
54 views

Series Summation

I have the series $$\sum_{k=1}^{N-1}\frac{1}{1-w_k} $$ where $w_k=e^{\frac{2\pi i k}{N}}$, how can I find the summation of this , another question related to this sum $$\sum_{k=1}^{N-1}\frac{1}{z-w_k} ...
0
votes
2answers
23 views

Series expansion with remaining $log n$

I'm studying the asymptotic behavior $(n \rightarrow \infty)$ of the following formula, where $k$ is a given constant. $$ \frac{1}{n^{k(k+1)/(2n)}(2kn−k(1+k) \ln n)^2}$$ I'm trying to do a series ...
1
vote
2answers
55 views

Partial fraction expansion two variables

How to expand $$\frac{y}{(x-y)(y-1)}$$ by partial fraction expansion.
6
votes
2answers
70 views

Evaluate the Sum $\sum_{i=0}^\infty \frac {i^N} {4^i}$ [duplicate]

$$\displaystyle\sum_{i=0}^\infty \frac {i^N} {4^i}$$ I'm supposed to evaluate this as I'm working through Data Structures and Algorithm Analysis in C++. I've solved similar problems, and after ...
1
vote
0answers
31 views

Bounding a sequence defined recursively

Let $\{x_n\}$ be a sequence of positive numbers and $\alpha \in (0,1)$. Let $y_0 := x_n$ and $$ y_k := (\alpha \,y_{k-1} + 1-\alpha) x_{n-k} $$ for $k=1,2,\dots,n-1$. Is it possible to give a sharp ...
0
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0answers
34 views

Linear Combinations of Irrational Numbers: An Analysis on Architecture

Under what condition(s) is $$ k_1\omega_1+\cdots + k_n\omega_n=c,$$ where $k_i\in\mathbb{R\setminus Q}$ and $\omega_i, c\in \mathbb{Q}$? I'm essentially trying to show that this is the case only so ...
1
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6answers
111 views

Does $\sum_{n=1}^\infty(-1)^{n}\frac{n^n}{n!}$ converge or diverge

$\sum_{n=1}^\infty(-1)^{n}\frac{n^n}{n!} $ I got that it diverges but I am not sure
0
votes
1answer
50 views

Sequence version of L'Hospital's Rule

Consider two sequences $A_n$ and $B_n$ such that $B_n$ is monotonically decreasing and both $A_n$ and $B_n$ tend to zero. Now let us consider the limits ...
0
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1answer
39 views

What is the function given by $\sum_{n=0}^\infty \binom{b+2n}{b+n} x^n$, where $b\ge 0$, $|x| <1$

For a nonnegative integer $b$, and $|x|<1$, what is the function given by the power series $$ \sum_{n=0}^\infty \binom{b+2n}{b+n} x^n. $$ For $b=0$, this post shows $$ \sum_{n=0}^\infty ...
0
votes
1answer
29 views

Taylor and geometric series

1) iF $f(x) = x^2 +x$, find the taylor series for f centered at a = 2. 2)what is the sum from 1 to infinity of $(.95)^n$ I got these questions wrong on my last test, and I'm not really sure how to ...
0
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2answers
50 views

Definition of metastability

I was reading Terence Tao's blog post on analogies between soft and hard analysis when I saw that the soft analysis statement "$x_n$ is convergent" corresponds to the hard analysis statement "$x_n$ is ...
2
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2answers
35 views

Given $x_1=1,x_2=2,x_{n+2}=3x_{n+1}-x_n\forall n\in\mathbb N$. Find $x_n$.

Given $x_1=1,x_2=2,x_{n+2}=3x_{n+1}-x_n\forall n\in\mathbb N$. Find $x_n$. I tried to find ways to telescope, but failed. Please help. Thank you.
0
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2answers
22 views

How can I construct such set?

Is it possible to construct a uncountable set $S\subset (0,1]$ which satisfies: $I_0$ - Every sequence $(x_{n})\subset S$ has the property $\sum_{n=1}^\infty x_{n}<\infty$ $I_1$ - Take any ...
1
vote
1answer
30 views

Solving ODE using frobenius method. 3 coefficients

I'm trying to learn frobenius method by solving some problems (ODEs). For example: $$xy''+(2x+1)y'+(x+1)y=0$$ Let $y=\sum\limits_{n=0}^\infty a_nx^{n+r}$. Then, I took derivatives and put into the ...
3
votes
7answers
124 views

How to prove that $1/n!$ is less than $1/n^2$?

I want to prove $$\sum_{n=0}^{\infty} \frac{1}{n!}$$ is a converging series. So I want to compare it with $\sum_{n=0}^{\infty} \frac{1}{n^2}$. I want to do direct comparison test. How to prove ...
11
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3answers
138 views

$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$

Is it possible to calculate the following infinite sum in a closed form? If yes, please point me to the right direction. $$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$$
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2answers
101 views
+500

Conjectural closed-form representations of sums, products or integrals

What are some examples of infinite sums, products or definite integrals that have conjectural closed form representations that were confirmed by numerical calculations up to whatever maximum precision ...
2
votes
1answer
42 views

Fourier series $\sum_{m=0}^\infty \frac{\cos (2m+1)x}{2m+1}$

Does anyone know the sum of Fourier series $$\sum_{m=0}^\infty \frac{\cos (2m+1)x}{2m+1}?$$ I tried WA; it does not return a function.
5
votes
2answers
94 views

Evaluating $\sum_{n=1}^{\infty} \frac{n}{e^{2 \pi n}-1}$ using the inverse Mellin transform

Inspired by this post, I tried to evaluate $\displaystyle \sum_{n=1}^{\infty} \frac{n}{e^{2 \pi n}-1} = \frac{1}{24} - \frac{1}{8 \pi}$ using the inverse Mellin transform. But my answer is twice as ...
2
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3answers
60 views

What type of Hypergeometric series is this?

I am trying to find a closed form for the series $$ \sum^\infty_{n=0} \frac{1}{n!} \frac{1}{n+1}(-z)^n {}_2F_2\left(m,n+1;\frac{1}{2},n+2; b z\right)$$ $m$ is a nonzero positive integer, and $b$, ...
2
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4answers
59 views

Approximation of alternating series $\sum_{n=1}^\infty a_n = 0.55 - (0.55)^3/3! + (0.55)^5/5! - (0.55)^7/7! + …$

$\sum_{n=1}^\infty a_n = 0.55 - (0.55)^3/3! + (0.55)^5/5! - (0.55)^7/7! + ...$ I am asked to find the no. of terms needed to approximate the partial sum to be within 0.0000001 from the convergent ...
0
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1answer
13 views

Geometric sequence, finding the first term using only the sum, the number of terms and value of one term.

In Geometric series: S = 56, a(2) = 16 and n = 3 S - sum, a(2) - second term, n - number of terms Is it possible to get a(2) and a(3) from here? (If yes, hints would be awesome) Thank You!
2
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0answers
52 views

About linear recurrence sequences

Let $\{a_n\}_{n=0}^\infty$,$\{b_n\}_{n=0}^\infty$,$\{c_n\}_{n=0}^\infty$ be three complex sequences and satisfy \begin{eqnarray*} &&\sum_{k=0}^2\alpha_ka_{n+k}=0,\\ ...
4
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2answers
39 views

Looking for a source of an infinite trigonometric summation and other such examples.

Question: If $x \neq 0$, then prove that $\displaystyle \sum_{n=1}^{\infty}\dfrac1{2^n} \tan\left(\dfrac{x}{2^n}\right) = \dfrac1{x} - \cot x.$ My answer: I proved this result by using the ...
0
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2answers
29 views

Whether an infinite series can be tested by integral test

I am asked whether the following infinite series can be proved to be convergent by integral test. $$\sum_{n=1}^\infty n e^{6 n}$$ so I integrate it $$\int_1^{\infty}\ n e^{6n}\, dn$$ and find it ...
1
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2answers
37 views

Values of a parameter $x$ in an infinite series that makes it converge

I am required to find the values of $x$ in the following infinite series, which cause the series to converge. $$\sum_{n=1}^\infty \frac{x^n}{\ln(n+1)}$$ I tried to use the ratio test, and found that ...
3
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1answer
66 views

effective way to get the integer sequence A181392 from oeis

the sequence A181392 are perfect squares and any digit in the sequence says "I am part of an integer in which you'll find d digits "d"" (see A108571, How can we call them? "digit-valid"?) How to get ...
3
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0answers
42 views

The Tribonacci constant and the Dragon

Let $x = \frac{\ln T}{\ln 2} = 0.879146\dots$ where $T$ is the tribonacci constant, then x solves the transcendental equation, $$4^x(2^x-1)=(2^x+1)$$ Let $x = \frac{\ln y}{\ln 2} = 1.523627\dots$ ...
10
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1answer
99 views

Inequality in a bounded real sequence

Prove or disprove that for any bounded real sequence $\{x_n\}_{n\in\mathbb{N}}$ there exist two distinct natural numbers $u,v$ such that: $$|x_u-x_v|\cdot|u-v|\leq 1.$$
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3answers
34 views

Sequence of the ratio of two successive terms of a sequence

If $(a_n)_{n\in N}$ is a strictly decreasing sequence of real number converging to $0$ and s.t. $\forall n\in N$, $0<a_n<1$, does the following limit: $$ \lim_{n}\frac{a_{n+1}}{a_n} $$ exists? ...
0
votes
2answers
31 views

Stochastic difference equation

I am a newbie in studying time series. Could anyone help solve the following problem: Consider the second-order stochastic difference equation: $y_t=1.5y_{t-1}-0.5y_{t-2}+\varepsilon_t$. Given ...
1
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0answers
47 views

Convergence tests [closed]

Test the series $k^n/ a^k$ , $a>1$ for convergence If are positive real numbers show that the sum $\sum \frac{(\ln k)^p}{k^q}$. State any test used. ...
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3answers
59 views

Comparison test for series $\sum_{n=1}^{\infty}\frac{n}{n^3 - 2n + 1}$

I am trying to prove the convergence of the series $\sum_{n=1}^{\infty}\frac{n}{n^3 -2n +1}$ with the simple comparison test. I know it can be done with other tests but this question came up in my ...
1
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2answers
54 views

Showing that $ \sum \limits_{m=1}^{n} b_m x_{m-n}~\to~ ab$ as $n~\to~\infty$

If $x_n ~\to ~a$ as $n~ \to~ \infty$ Does: $ \sum \limits_{m=1}^{n} b_m x_{n-m}~\to~ ab$ as $n~\to~\infty$? $b_m ~\geq~0$ and $ b~\equiv~ \sum \limits_{m=1}^{\infty} b_m < \infty$ My attempt: ...
1
vote
3answers
51 views

Limit as N goes to Infinity

Consider this limit: $$\lim_{n\rightarrow\infty} \left( 1+\frac{1}{n} \right) ^{n^2} = x$$ I thought the way to solve this for $x$ was to reduce it using the fact that as $n \rightarrow \infty$, ...
13
votes
2answers
184 views

Closed form for $\sum_{n=1}^\infty\frac{1}{2^n\left(1+\sqrt[2^n]{2}\right)}$

Here is another infinite sum I need you help with: $$\sum_{n=1}^\infty\frac{1}{2^n\left(1+\sqrt[2^n]{2}\right)}.$$ I was told it could be represented in terms of elementary functions and integers.
3
votes
1answer
59 views

Proof the following trig series

Prove that $$\frac{ \sin x}{ \cos x}+\frac{\sin2x}{\cos^{2}x}+\frac{\sin3x}{\cos^{3}x}+\cdots+\frac{\sin nx}{\cos^{n}x}=\cot x-\frac{\cos(n+1)x}{\sin x \cos^{n}x}$$ I am not necessarily looking for a ...
1
vote
0answers
43 views

Is zero a cluster point of $n\sin n$?

I have seen somewhere that if $0\le \alpha<1$, then zero is a cluster point of the sequence $n^\alpha \sin n, n=1,2,\cdots$. My question is what if $\alpha=1$? Or $\alpha>1$?
2
votes
2answers
52 views

Holomorphic series with its real part positive $f(z)=1+\sum_{n=1}^\infty a_n z^n$

Let $$f(z)=1+\sum_{n=1}^\infty a_n z^n,$$ $f \in H(B(0,1))$, and $\operatorname{Re} f(z)\ge 0$, $\forall z \in B(0,1) $. Prove: (1) $| a_n | \le2$; (2) $|a_1^2-a_2| \le 2, ...

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