Recurrence relations, convergence tests, identifying sequences
0
votes
2answers
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
votes
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
votes
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
vote
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
votes
1answer
36 views
5
votes
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:
$$
\begin{align}
x_0 & = a \\
x_1 & = x_0 + \frac{1}{1}(x_0 + x_0(c - 1)) \\
x_2 & = x_1 + \frac{1}{2}(b + x_1(c - 1)) \\
x_3 & = x_2 + \frac{1}{3}(b + x_2(c - 1)) ...
1
vote
3answers
53 views
Showing that $\sum_{n=3}^\infty\frac1{n(\log n)(\log\log n)}$ diverges
I know that the series diverge, I'm just having hard time showing it.
$$\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
votes
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
vote
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
votes
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
$$
\sum_{n=0}^\infty ...
0
votes
1answer
29 views
Taylor and geometric series
1) iF $f(x) = x^2 +x$, find the taylor series for f centered at a = 2.
2)what is the sum from 1 to infinity of $(.95)^n$
I got these questions wrong on my last test, and I'm not really sure how to ...
0
votes
2answers
50 views
Definition of metastability
I was reading Terence Tao's blog post on analogies between soft and hard analysis when I saw that the soft analysis statement "$x_n$ is convergent" corresponds to the hard analysis statement "$x_n$ is ...
2
votes
2answers
35 views
Given $x_1=1,x_2=2,x_{n+2}=3x_{n+1}-x_n\forall n\in\mathbb N$. Find $x_n$.
Given $x_1=1,x_2=2,x_{n+2}=3x_{n+1}-x_n\forall n\in\mathbb N$. Find $x_n$.
I tried to find ways to telescope, but failed. Please help. Thank you.
0
votes
2answers
22 views
How can I construct such set?
Is it possible to construct a uncountable set $S\subset (0,1]$ which satisfies:
$I_0$ - Every sequence $(x_{n})\subset S$ has the property $\sum_{n=1}^\infty x_{n}<\infty$
$I_1$ - Take any ...
1
vote
1answer
30 views
Solving ODE using frobenius method. 3 coefficients
I'm trying to learn frobenius method by solving some problems (ODEs).
For example:
$$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!}$$ is a converging series. So I want to compare it with $\sum_{n=0}^{\infty} \frac{1}{n^2}$. I want to do direct comparison test.
How to prove ...
11
votes
3answers
138 views
$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$
Is it possible to calculate the following infinite sum in a closed form? If yes, please point me to the right direction.
$$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$$
17
votes
2answers
101 views
+500
Conjectural closed-form representations of sums, products or integrals
What are some examples of infinite sums, products or definite integrals that have conjectural closed form representations that were confirmed by numerical calculations up to whatever maximum precision ...
2
votes
1answer
42 views
Fourier series $\sum_{m=0}^\infty \frac{\cos (2m+1)x}{2m+1}$
Does anyone know the sum of Fourier series $$\sum_{m=0}^\infty \frac{\cos (2m+1)x}{2m+1}?$$
I tried WA; it does not return a function.
5
votes
2answers
94 views
Evaluating $\sum_{n=1}^{\infty} \frac{n}{e^{2 \pi n}-1}$ using the inverse Mellin transform
Inspired by this post, I tried to evaluate $\displaystyle \sum_{n=1}^{\infty} \frac{n}{e^{2 \pi n}-1} = \frac{1}{24} - \frac{1}{8 \pi}$ using the inverse Mellin transform. But my answer is twice as ...
2
votes
3answers
60 views
What type of Hypergeometric series is this?
I am trying to find a closed form for the series
$$ \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)$$
$m$ is a nonzero positive integer, and $b$, ...
2
votes
4answers
59 views
Approximation of alternating series $\sum_{n=1}^\infty a_n = 0.55 - (0.55)^3/3! + (0.55)^5/5! - (0.55)^7/7! + …$
$\sum_{n=1}^\infty a_n = 0.55 - (0.55)^3/3! + (0.55)^5/5! - (0.55)^7/7! + ...$
I am asked to find the no. of terms needed to approximate the partial sum to be within 0.0000001 from the convergent ...
0
votes
1answer
13 views
Geometric sequence, finding the first term using only the sum, the number of terms and value of one term.
In Geometric series: S = 56, a(2) = 16 and n = 3
S - sum, a(2) - second term, n - number of terms
Is it possible to get a(2) and a(3) from here? (If yes, hints would be awesome)
Thank You!
2
votes
0answers
52 views
About linear recurrence sequences
Let $\{a_n\}_{n=0}^\infty$,$\{b_n\}_{n=0}^\infty$,$\{c_n\}_{n=0}^\infty$ be three complex sequences and satisfy
\begin{eqnarray*}
&&\sum_{k=0}^2\alpha_ka_{n+k}=0,\\
...
4
votes
2answers
39 views
Looking for a source of an infinite trigonometric summation and other such examples.
Question:
If $x \neq 0$, then prove that $\displaystyle \sum_{n=1}^{\infty}\dfrac1{2^n} \tan\left(\dfrac{x}{2^n}\right) = \dfrac1{x} - \cot x.$
My answer:
I proved this result by using the ...
0
votes
2answers
29 views
Whether an infinite series can be tested by integral test
I am asked whether the following infinite series can be proved to be convergent by integral test.
$$\sum_{n=1}^\infty n e^{6 n}$$
so I integrate it
$$\int_1^{\infty}\ n e^{6n}\, dn$$
and find it ...
1
vote
2answers
37 views
Values of a parameter $x$ in an infinite series that makes it converge
I am required to find the values of $x$ in the following infinite series, which cause the series to converge.
$$\sum_{n=1}^\infty \frac{x^n}{\ln(n+1)}$$
I tried to use the ratio test, and found that ...
3
votes
1answer
66 views
effective way to get the integer sequence A181392 from oeis
the sequence A181392 are perfect squares and any digit in the sequence says
"I am part of an integer in which you'll find d digits "d"" (see A108571, How can we call them? "digit-valid"?)
How to get ...
3
votes
0answers
42 views
The Tribonacci constant and the Dragon
Let $x = \frac{\ln T}{\ln 2} = 0.879146\dots$ where $T$ is the tribonacci constant, then x solves the transcendental equation,
$$4^x(2^x-1)=(2^x+1)$$
Let $x = \frac{\ln y}{\ln 2} = 1.523627\dots$ ...
10
votes
1answer
99 views
Inequality in a bounded real sequence
Prove or disprove that for any bounded real sequence $\{x_n\}_{n\in\mathbb{N}}$ there exist two distinct natural numbers $u,v$ such that:
$$|x_u-x_v|\cdot|u-v|\leq 1.$$
0
votes
3answers
34 views
Sequence of the ratio of two successive terms of a sequence
If $(a_n)_{n\in N}$ is a strictly decreasing sequence of real number converging to $0$ and s.t. $\forall n\in N$, $0<a_n<1$, does the following limit:
$$
\lim_{n}\frac{a_{n+1}}{a_n}
$$
exists?
...
0
votes
2answers
31 views
Stochastic difference equation
I am a newbie in studying time series. Could anyone help solve the following problem:
Consider the second-order stochastic difference equation: $y_t=1.5y_{t-1}-0.5y_{t-2}+\varepsilon_t$. Given ...
1
vote
0answers
47 views
Convergence tests [closed]
Test the series $k^n/ a^k$ , $a>1$ for convergence
If are positive real numbers show that the sum $\sum \frac{(\ln k)^p}{k^q}$.
State any test used. ...
1
vote
3answers
59 views
Comparison test for series $\sum_{n=1}^{\infty}\frac{n}{n^3 - 2n + 1}$
I am trying to prove the convergence of the series $\sum_{n=1}^{\infty}\frac{n}{n^3 -2n +1}$ with the simple comparison test. I know it can be done with other tests but this question came up in my ...
1
vote
2answers
54 views
Showing that $ \sum \limits_{m=1}^{n} b_m x_{m-n}~\to~ ab$ as $n~\to~\infty$
If $x_n ~\to ~a$ as $n~ \to~ \infty$
Does:
$ \sum \limits_{m=1}^{n} b_m x_{n-m}~\to~ ab$ as $n~\to~\infty$?
$b_m ~\geq~0$ and $ b~\equiv~ \sum \limits_{m=1}^{\infty} b_m < \infty$
My attempt:
...
1
vote
3answers
51 views
Limit as N goes to Infinity
Consider this limit: $$\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, ...
