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

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0
votes
2answers
25 views

Prove convergence of sequence

A sequence $\{x_n\}$ satisfies $$|x_{n+1}-x_n| \leq \alpha|x_n-x_{n-1} | $$ for $n=2,3,4,\ldots$ and the number $ 0<\alpha<1$ fixed. Prove $\{x_n\}$ is a convergent sequence.
0
votes
2answers
24 views

Is my procedure correct about sequences?

Let $\alpha\in(0,2)$, and the sequence $$x_{n+1}=\alpha x_n +(1+\alpha)x_{n-1} \quad \forall n\geq 1$$ Find the limit in terms of $\alpha$, $x_0$ and $x_1$. Check my work. If $\alpha=1$, ...
0
votes
1answer
30 views

If $\{a_{n_k}\}$ is a subset of $\{a_n\}$, $\lim_{k\to\infty} a_{n_k }= \lim_{n\to\infty} a_n\ $

Let $\{a_n\}$ be a sequence and L a real number such that $\lim_{n\to\infty} a_n = L$ Prove that if $\{a_{n_k}\}$ is any subsequence of $\{a_n\}$, then $\lim_{k\to\infty} a_{n_k} = L $ I have ...
0
votes
1answer
28 views

Prove that $\{a_n\}$ is bounded [duplicate]

Let $\{a_n\}$ be a sequence an $L$ a real number such that $\lim_{n\to\infty} a_n = L$ Prove that $\{a_n\}$ is bounded This reminds me of the bounded monotone convergence theorem (BMCT) but in ...
3
votes
3answers
72 views

Determining the value to which the sequence $a_n=\frac{n!}{n^n}$ converges.

How can it be deduced that the sequence $a_n=\dfrac{n!}{n^n}$ converges to $0$? I can reasonably infer this to be true, because I see the pattern as $n$ approaches larger values, but I am unsure of ...
-3
votes
1answer
32 views

Prove that $f(x)$ is continuous at $a$

Prove that f(x) is continuous at a if and only if for all sequences sequence $\{a_n\}$ with $\lim_{n\to\infty} a_n = a$, $\lim_{n\to\infty} f(a_n) = f(a)$ I have no idea how to start this. Should I ...
1
vote
1answer
41 views

Sign fluctuation in the harmonic series (sum)

Lets begin with a few simple rules that we know. -$\sum_{k=1}^\infty \frac{1}{n}=\infty$ -$\sum_{k=1}^\infty \frac{\color{red}{(}-1\color{red}{)}^{n-1}}{n}= \ln2 $ Going off of this knowledge, we ...
0
votes
1answer
34 views

Evaluating the convergence of the sequence $\{a_n\}=\frac{(-1)^{n-1}n}{n^2+2}$.

Set the sequence $a_n$ such that $\{a_n\}=\dfrac{(-1)^{n-1}n}{n^2+2}$. If $|a_n|$ converges (only to $0$, it would seem; correct me if I'm wrong), then $a_n$ must too converge, both to some value $L = ...
2
votes
2answers
27 views

Finding the first three terms of a geometric sequence, without the first term or common ratio.

Given a geometric sequence where the $5$th term $= 162$ and the $8$th term $= -4374$, determine the first three terms of the sequence. I am unclear how to do this without being given the first ...
-1
votes
0answers
44 views

Is this $\sum_{k=1,\theta \in \mathbb{R}}^{k=n}\frac{\cos k\theta}{k} $ alternating series for all values of $\theta$?

I have tried to do other form of alternating series I got this: $$\sum_{k=1,\theta \in \mathbb{R}}^{k=n}\frac{\cos k\theta}{k} $$ Can I say that the above series is alternating series for all ...
1
vote
1answer
13 views

Positive convergent sequence. Existence of another positive convergent sequence with same limit and larger elements

I have a positive sequence which converges to zero, i.e. $a_k \geq 0 \;, \forall k \in \mathbb{N}$ and $\lim_{k\rightarrow \infty} a_k = 0$. Does there exist another sequence $b_k$ with the property ...
0
votes
2answers
66 views

How do I evaluate this:$\sum_{n=1}^{\infty}\frac{1}{n²}(e^x −1 −\frac{x}{1!} −\frac{x²}{2!}−\cdots\frac{x^n}{n!})$?

How do i evaluate this sum :$$\sum_{n=1}^{\infty}\frac{1}{n²}(e^x −1 −\frac{x}{1!} −\frac{x²}{2!}−\cdots\frac{x^n}{n!})$$ Note: I 'd surprised if it is convergent Thank you for any help.
3
votes
2answers
51 views

If $\{a_n\}$ converges to $A$, then $\{(a_1\cdots a_n)^{1/n}\}$ converges to $A$

Prove that this sequence converges. I can't do it. Let $\{a_n\}$ be a sequence of positive real numbers that converges to a number $A$. Prove that $\{(a_1\cdots a_n)^{1/n}\}$ converges to $A$.
1
vote
1answer
55 views

Is my proof correct about sequences?

Suppose that $\{ a_n\}_n$ is a sequence of real numbers such that $$ (a_{n+1}-a_n) \rightarrow a, \text{ if } \ n \rightarrow \infty. $$ Prove that $$ \frac{a_n}{n} \rightarrow a \, \text{ if } \ ...
1
vote
1answer
48 views

Series convergence proof review (Baby Rudin)

Ch.3 #7. Prove that the convergence of $\sum a_n$ implies the convergence of $$\sum \frac{\sqrt{a_n}}{n},$$ if $a_n \geq 0$. My attempt. If $\sum a_n$ is convergent, then by the root test, ...
0
votes
0answers
22 views

Confused between cyclic sum and symmetric sums.

four variables $a, b, c, d$ are given, what is the symmetric and cyclic sum? I thought: $$\sum_{cyc} ab = ab + ac + ad + bc + bd + cd$$ And $$\sum_{sym} ab = 2(ab + ac + ad + bc + bc + ...
0
votes
1answer
37 views

Determine whether $\sum \frac{1}{n^3 \ln(n^4+9)}$ converges

For the series $$\sum_{n=2}^{\infty}\dfrac{1}{n^3 \ln(n^4+9)},$$ I was thinking of using the limit comparison test with $1/n^3$?
1
vote
1answer
45 views

solution for $\prod_{n=1}^{N}\left ( \frac{1}{1+e^{n} } \right )$

Is there a closed form for $$\prod_{n=1}^{N}\left ( \frac{1}{1+e^{n} } \right )?$$ I'm looking for the solution of $$\prod_{n=1}^{N}\left (1-\left ( \frac{1}{1+e^{-n} } \right )\right )$$ Thanks a ...
1
vote
5answers
55 views

Determine whether $\sum \frac{2^n + n^2 3^n}{6^n}$ converges

For the series $$\sum_{n=1}^{\infty}\dfrac{2^n+n^23^n}{6^n},$$ I was thinking of using the root test? so then I would get $(2+n^2/n+3)/6$ but how do I find the limit of this?
1
vote
4answers
66 views

Convergence of $\sum_{n=1}^{\infty}\frac{1}{(3n+8)!}$

Determine whether or not this series converges or diverges $$\sum_{n=1}^{\infty}\dfrac{1}{(3n+8)!}$$ My attempt: I used the ratio test and ended up having ...
4
votes
1answer
58 views

Find $\sum_{n=1}^{\infty}\ln(1+\frac{1}{n})\ln(1+\frac{1}{2n})\ln(1+\frac{1}{2n+1})$

Here is another problem from IMC competition. Solution given there: solutions, is totally taken out of blue for me. What is more I discovered the relationship ...
0
votes
1answer
15 views

Finding probabilities from probabilty generating function

Given that I have a probability generating function for $Q$ given by $\dfrac{4s^{2}}{9-3s-2s^{2}}$, I want to find $P(Q = n)$ for $n \geq 2$. I understand that I could actually use the definition of ...
1
vote
1answer
36 views

Sum of function series is continuous

Given the sum $\sum_{n=1}^{\infty}n^5(\frac{x}{x+2})^n$ I want to show that this sum is a continuous function in $[0,10]$. What is the proccess I need to go through when doing so? Do I first need to ...
0
votes
0answers
19 views

Series expansion of reciprocal function of Generalized Exponential Integral

Generalized Exponential Integral of order p has a series expansion http://dlmf.nist.gov/8.19.10 Is there a series expansion of the reciprocal function ?
0
votes
0answers
23 views

Regularization of a (divergent) cosine series

What would be a suitable regularized value for the following divergent series: $$ S(y) = \sum_{k=1}^{\infty} \cos(k y) \quad y \in R\\ $$ By way of added context, this series arises from a formal ...
1
vote
1answer
30 views

Uniform convergency of a two sequence of functions

For, $n\ge 1$ let, $g_n(x)=\sin^2(x+1/n)$ , $x\in [0,\infty)$ and $f_n(x)=\int_0^xg_n(t)\,dt.$ Then, which is(/are) correct ? (A) $\{f_n(x)\} $ converges pointwise to a function $f$ on $[0,\infty)$ ...
2
votes
4answers
72 views

$0,1,0,1,0,1$… has only $2$ limit points

Prove that the sequence $0,1,0,1,0,1$... has only $2$ limit points : $0$ and $1$. To be frank, I know the solution to the above particular problem. What I am interested is in knowing a general ...
2
votes
2answers
29 views

Finding error in reasoning about geometric series sum

A supposedly fast method to find the sum of a geometric series is the following one. Let $$S = \sum_{n = 0}^{+\infty} q^n$$ then $$S = 1 + q\left[\sum_{n = 0}^{+\infty} q^n\right] = 1 + qS.$$ Hence ...
4
votes
1answer
89 views

Proving a convergence relationship between two sequences

Let $a_{n}$ a sequence of real numbers. Let $\sigma_n= \frac{a_1+a_2+...+a_n}{n}$. Suppose that $\lim_{n\to \infty} \sigma_n=A.$ Prove that $$\lim_{n \to \infty}\frac{1}{\log n} ...
4
votes
3answers
2k views

Expressing the infinite sum 1 + 22 + 333 + 4444 + …

I would like to express $$1+22+333+4444+\cdots$$ using $\Sigma$ notation, and have no clue where to start. After $999999999$, comes 10 $0$s, then 11 $1$s.
4
votes
1answer
122 views

Find all values such that the series converge:

Find all the values of $p\in\mathbb R$ such that the following series converge: $$\sum_{k=2}^\infty (\log k)^{p\log k}$$ I would like hints only. I've tried using the exponential function ...
-3
votes
1answer
39 views

What is value of k in next hermonic serie? [on hold]

I want to know the value of k such that $\sum_{n=1}^{k}\frac{1}{n} \geq 8000$. Thanks.
0
votes
1answer
41 views

Prove this sequence is bounded

This is not any exercise on itself, but I was reading a proof in which a sequence similar to this appeared: $$a_n=n\lambda^{n-1}, \quad|\lambda|<1$$ In essence. Then I came across the assertion, ...
6
votes
1answer
75 views

Distribution of the sum reciprocal of primes $\le 1$

$$\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{43}+\cdots \le 1 $$ This is an interesting infinite summation. This is very closely resembling my other problem with has to do with the distribution of ...
3
votes
1answer
53 views

Strategies For Summing Harmonic Numbers

Lately, I have found several interesting problems involving Harmonic numbers such as \begin{equation*}\sum_{n=1}^{\infty}\frac{H_n^{(2)}}{n^2}=\frac{7\pi^4}{360}\end{equation*} I am not familiar with ...
3
votes
1answer
33 views

Identities For Generalized Harmonic Number

I have been searching for identities involving generalized harmonic numbers \begin{equation*}H_n^{(p)}=\sum_{k=1}^{n}\frac{1}{k^p}\end{equation*} I found several identities in terms of $H_n^{(1)}$, ...
0
votes
1answer
32 views

How do I find if a series of integers is (somehow) regular

[DISCLAMIER]: please be patient and low profile: no math background here :) I have a database of transactions, i.e. a list of purchases made by different customers. What I have to figure out is ...
9
votes
1answer
117 views

Computing $\sum _{k=1}^{\infty } \frac{\Gamma \left(\frac{k}{2}+1\right)}{k^2 \Gamma \left(\frac{k}{2}+\frac{3}{2}\right)}$ in closed form

What tools other than beta function you might like to use here? $$\sum _{k=1}^{\infty } \frac{\displaystyle \Gamma \left(\frac{k}{2}+1\right)}{\displaystyle k^2 \Gamma ...
-4
votes
4answers
66 views

Find the limit $\lim_{n \to \infty} \sum_{k=1}^n \frac{1}{k+n}$ [on hold]

I need help finding the following limit. $$\lim_{n \to \infty} \sum_{k=1}^n \frac{1}{k+n}$$
-9
votes
1answer
46 views

I need to find an nth term for a fairly random sequence [on hold]

I need an nth term of the sequence 6,2,0,3,1,8,10,5,6,6,4,5,4,6 there is no method for finding the next numbers, all i need is an nth term which will work. thanks :D
2
votes
1answer
42 views

Recurrence relation involving infinite sequences.

How in general one would solve an infinite series recurrence relations? For instance, I am interested to solve the following: \begin{equation} \sum_{n=0}^{\infty} (-1)^{n} F(n)\{1-(\alpha n ...
6
votes
1answer
99 views

Does $ \sum_{(m,n) \neq (0,0)} \frac{(-1)^{m+n}}{m^2 + n^2} $ have an exact value?

I am looking for an $\mathbb{Z}[i]$ analogue of the alternating harmonic series: $L(1,\chi)=\sum_{n=0}^\infty (-1)^n \frac{1}{n} = \frac{\pi}{4}$. If we try adding the reciprocals of the Gaussian ...
0
votes
0answers
12 views

How to optimize the repartition of samples in weighted channels?

This is more like an applied mathematics question, so my apologies if I am at the wrong place. Let S(n) be an infinite sequence of real numbers strictly growing from 0 to 1 (asymptotically). Let P be ...
0
votes
4answers
56 views

Show $ \lim_{n\rightarrow \infty} 2^{-1/\sqrt{n}}=1$

I am tasked with proving the following limit: $$ \lim_{n\rightarrow \infty} 2^{-1/\sqrt{n}}=1$$ using the definition of the limit. I think I have done so correctly. I was hoping to have someone ...
7
votes
2answers
84 views

Does the sum of the reciprocals of composites that are $ \le $ 1

The sum itself: $$ \frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{12}+\frac{1}{14}+ \frac{1}{15}+ \frac{1}{39}... \le 1 $$ These are all sums of reciprocals of composites that ...
2
votes
2answers
170 views

Help with Convergence of a series with sin and log

I tried to figured it out if the follwing series converges or not $$\sum_{n=1}^{\infty} \frac{\sin(\frac{1}{n})}{\ln^2n}\ (-1)^{n}$$ I tried to show that $\sin(\frac{1}{n})$ is a monotonic but I'm ...
0
votes
1answer
30 views

Discuss the convergence of $\sum a^n/(x^n+a^n)$?

I fail to understand how any of the tests I know will work on this. I tried the D'Alembert's Ratio test but I don't understand how it will limit to something.
1
vote
5answers
88 views

Proving that $\forall x>0$, $\lim\limits_{n\to\infty}x^{1/n}=1$

I was reading sequences in Terence Tao Analysis book and I came across the question: Prove that $\forall x>0$, $\lim\limits_{n\to\infty}x^{1/n}=1$ In the hint it says that ... you may ...
1
vote
1answer
20 views

Non-Cauchy products for series?

Let $\sum a_n$ and $\sum b_n$ be two absolutely convergent series. My question is Is it possible to take the product of the two series $(\sum a_n)(\sum b_n)$ and get a result different from the ...
1
vote
3answers
26 views

Should I use the comparison test for the following series?

given the following series $\sum_{k=0}^\infty \frac{\sin(2k)}{1+2^k}$ I'm supposed to determine whether it converges or diverges. Am I supposed to use the comparison test for this? My guess would ...