1
vote
1answer
25 views

Compute infinite sum of a arithmetico-geometric series $\sum_{i=0}^{\infty} \frac{i}{2^i}$ [duplicate]

I am trying to compute the sum $\sum_{i=0}^{\infty} \frac{i}{2^i}$ which I know should be equal to $2$, but I cannot prove it. If I am not mistaken, it should be a arithmetico-geometric series ...
2
votes
1answer
166 views

Does series converge or not?

$$\sum_{n=1}^\infty~\left|\frac{\cos2^n}{n}\right|$$ I just confused what to do.
0
votes
3answers
40 views

Alternating p series. given that summation

Given that $$\sum_{k=1}^\infty{\frac{1}{k^2}} = \frac{\pi^2}{6}\ $$ Show that $$\sum_{k=1}^\infty{\frac{(-1)^{k+1}}{k^2}} = \frac{\pi^2}{12}\ $$
0
votes
2answers
27 views

A question about limsup and limif

Could you please help me understand this question: Suppose $a_n$ is bounded sequence and $A<\liminf a_n$, $B>\limsup a_n$. Prove : $A<a_n<B$ for all n>N. It seems to me to simple to be ...
0
votes
2answers
39 views

Calculate the following sequence $\sum_{n=0}^{+\infty }\left ( -\dfrac{1}{4\alpha } \right )^{n}\dfrac{ (2n)!}{n!},\; \alpha >0$

Calculate the following sequence $$\sum_{n=0}^{+\infty }\left ( -\dfrac{1}{4\alpha } \right )^{n}\dfrac{ (2n)!}{n!},\; \alpha >0$$
1
vote
1answer
33 views

Sequence bounds and limit

I'm doing the following exercise. Given the sequence \begin{cases} a_{n+1} = {n + 8\over4n + 1}*a_n & n=0, 1, 2 \\ a_{0} = 1 \end{cases} Find if the sequence is definitely decreasing/increasing. ...
0
votes
3answers
33 views

Convergance of a sequence [on hold]

Prove that the sequence $(a_n)$ converges, where$$a_n=\frac {3+n+4{n^2}}{1-n+3{n^2}}$$ for all $n\ge1$
0
votes
1answer
11 views

Multiplying non-decreasing sequences

Let $(a_n)$ and $(b_n)$ be non-decreasing sequences of positive terms (i.e. $a_n\gt0$ and $b_n\gt0$ for all $n\ge1$). Prove that the sequence $(c_n)$ is non-decreasing, where $c_n=a_nb_n$ for all ...
2
votes
1answer
31 views

Problem with Sequences..

I'm having trouble with a homework question, and cant see where im going wrong. The question is as follows: Let $x\gt1$ and let $a_n = 1+\frac 1x+\frac 1{x^2} + \cdots + \frac 1{x^n}$ for all ...
1
vote
2answers
30 views

suming an infinite series

the question involves person A eating half a loaf of bread. then person B eats half of the half left over. then person A eats half of whats left over... etc. I defined the series of person A as ...
1
vote
3answers
52 views

Finding the limit of a sequence by diagonalising a matrix

Consider the sequence described by: $\frac11 , \frac32 , \frac75 , ... ,\frac {a_{n}}{b_{n}}$ where $ a_{n+1} = a_n +2b_n $ and $b_{n+1} = a_n+b_n$ Find a matrix $A$ such that ...
2
votes
2answers
53 views

How to prove convergence of $a_n$ if $(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$?

Could you give me some hint how to conclude convergence of $a_n$ from this feature : $$(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$$ From $$(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$$ we may conclude that ...
0
votes
0answers
25 views

Construct a Converging Series from the Following

This is more of a request for advice than a request for solution. Last night we were given the following and nobody figured it out in the time given (about 5 minutes). I think this is a problem many ...
0
votes
4answers
42 views

Determine if the given sequence converges or diverges

Let $(x_n)$ be a sequence defined as $x_n = \frac{1}{n} \sum_{j=1}^{n} \frac{j+1}{j^2}$ . We want to know if $(x_n)$ converges. The trouble I am having here is that the sum depends on $n$. We know the ...
1
vote
3answers
71 views

Determine if $\sum\limits_{n=1}^{\infty}(1+\dfrac{2}{n})^n$ converges or diverges

I have an infinite series $\sum\limits_{n=1}^{\infty}(1+\frac{2}{n})^n$. I need to show if it converges or diverges using any test. I've tried applying all of the tests that I know, and hit dead ...
0
votes
2answers
40 views

A question about using Squeeze Theorem to solve theoretical convergence question

Could you give me some hint how to deal with this question: Suppose $a_n\le b_n \le c_n$ for almost all n, $b_n\to L$, $c_n-a_n\to 0$. Prove: $a_n \to L,b_n \to L$. Well, if $a_n\to a, b_n \to b$ ...
2
votes
2answers
118 views

How to calculate $\lim_{n \to \infty} \frac 1{3n} +\frac 1{3n+1}+\cdots+\frac 1{4n}$?

Could you please help me calculate this limit: $\lim_{n \to \infty} \frac 1{3n} +\frac 1{3n+1}+\cdots+\frac 1{4n}$. My best try is : $\lim_{n \to \infty} \frac 1{3n} +\frac 1{3n+1}+\cdots+\frac ...
1
vote
3answers
29 views

Don't know why this power series representation is wrong…

I've run into something confusing. The problem is that I have to find the power series representation of $g(x)$ using the given $f(x)$, specifically $g(x) = \ln(1 - 3x)$ using $f(x) = \frac{1}{1-x}$. ...
0
votes
1answer
33 views

Convergence and sum of geometric series (e^(3-2n)) as n goes from 2 to infinity

I have simplified the expression to: (e^3 / e^2n) This particular question asks to answer whether or not the series converges by virtue of |common ratio| < 1 alone, without using any other tests ...
1
vote
1answer
39 views

How to prove : If $a_{2n},a_{2n-1},a_{7n}$ converges than $a_n$ also converges.

Could you give me some hint how to prove this statement: If $a_{2n},a_{2n-1},a_{7n}$ converges than $a_n$ also converges. I think, obviously wrong, that if $a_{2n}\to a$ and $a_{2n-1}\to b$ than ...
1
vote
1answer
51 views

Is this sum converges or not?

$$\int_{n=2}^\infty \frac{\arctan\Big((-1)^nn^2\Big)}{n\ln^3n}$$ i will be glad if anyone can help me. I tried comparing it to the sum of $\Sigma_{n=2}^{\infty}\frac{1}{nlnn}$ and i said the integral ...
0
votes
1answer
24 views

Question about bounded sequence with two sub-sequential limits.

Could you please give me some hint how to deal with this question. Suppose $(a_n)$ is bounded sequence with 2 sub-sequential limits. Prove : there are real numbers A and B that ...
2
votes
2answers
84 views

If $\sum a_n$ converges then $\sum (-1)^n \frac {a_n}{1+a_n^2}$ converges?

Could you please give me some hint how to deal with this question: If $\sum a_n$ converges, does this necessarily mean that $\sum (-1)^n \frac {a_n}{1+a_n^2}$ must converge also ? Thanks.
2
votes
2answers
29 views

Help with Maclaurin Series

I am working on finding a Maclaurin series for this function. $$f(x) =x^6e^{x^7}$$ So I think I have to evaulate the above function based on a Maclaurin series for $e^x$ = $\sum_{n=0}^\infty ...
1
vote
2answers
76 views

Show that $\sum\limits_{n=1}^\infty\dfrac{2n^2-1}{3n^5+2n+1}$ converges or diverges

I'm working with some infinite series problems and I have to show that the series $\sum\limits_{n=1}^\infty\dfrac{2n^2-1}{3n^5+2n+1}$ converges or diverges. I don't have a lot of experience doing ...
3
votes
2answers
142 views

Does $\sin(\sin(\sin\cdots(\sin1)\cdots) \rightarrow 0 $?

Stuck on homework problem (not this), if I can prove as a lemma that the sequence $$\sin(\sin(\sin\cdots(\sin1)\cdots) \rightarrow 0 $$ then I'm done. It's monotonic and decreasing and bounded by 0 ...
1
vote
1answer
31 views

Find a general formula for x_k

The sequence $x_k$... is defined by $x_0 = 0, x_1 = 2$, and $x_{k+2} = 6x_{k+1}−13x_k$ for $k≥0$. Find a general formula for $x_k$. I actually came here because I found a solution on here for a ...
0
votes
0answers
17 views

The summation of product of factorials

So the question is $\sum\limits_{x=0}^n \frac{(\beta+n-x)! (\alpha+x)!}{x!(n-x)!}$. I got the following result from mathematica yet I don't know how to prove it. Can anyone give me some help?
0
votes
0answers
30 views

Superlinearly convergent

A sequence $\{p_n\}$ is said to be superlinearly convergent to $p$ if $$\lim_{n\to \infty}{\frac{|p_{n+1}-p|}{|p_n-p|}}=0$$ a. Show that if $p_n\to p$ of order $\alpha$ for $\alpha>1$, then ...
0
votes
0answers
32 views

Question on Morse inequalities

I want to understand why: if i have then $(4.1)$ is formal : it means that please help me Thank you EDIT1: $(4.1)$ tel us that $\displaystyle\sum_{q=0}^{\infty} (M_q-\beta_q)t^q=(1+t)Q(t)$ ...
0
votes
0answers
34 views

Short question on power series

After applying ratio test, the result L = x^2/4, n approaches infinity. So when you're trying to find the radius of convergence of: $$x^2/4<1$$ is it $$x<2?$$ Therefore radius is 2? I want ...
1
vote
2answers
52 views

$\sqrt[\infty]{\infty^2}$ in limit of series using root test

I'm trying to solve a problem to show if the infinite series $\sum\limits_{k=1}^{\infty}\dfrac{k^2}{2^k}$ converges or diverges using the root test. When put in limit form, I got ...
1
vote
2answers
41 views

Infinite series: which test

I'm having troubling deciding which test to use for this: $$\sum_{n=1}^\infty (-1)^n\arctan\left(\frac{\ln(n!)}{n+4^n}\right)$$ I tried altnerating test and ratio test but I couldn't get an answer. ...
4
votes
2answers
126 views

Infinite Series with factorial

I'm having trouble manipulating the function of this series which has factorials to show that it converges or diverges using the ratio test. The series is ...
1
vote
1answer
61 views

How to prove it?

Let $y_0\geqslant 2$, $y_n=y_{n-1}^2-2$, $n\in\mathbb{N}_+$, set $\displaystyle S_n=\sum_{k=0}^{n}\frac{1}{y_0\cdots y_k}$, how to prove $$\lim_{n\to\infty}S_n=\frac{y_0-\sqrt{y_0^2-4}}{2}.$$ Do you ...
1
vote
1answer
35 views

What is the main defferences between nets and ordinary sequences

I know that there are many results in metric spaces (or first-countable topological spaces) can be describe in the language of sequences but these results might not be true in general topological ...
1
vote
2answers
35 views

Sequence with an infinite amount of limit points

Find a sequence which has an infinite amount of limit points. I was thinking about using the bijective pairing function $\langle\cdot,\cdot\rangle:\Bbb N\times\Bbb N\to\Bbb N,\langle ...
0
votes
1answer
21 views

Determining value of infinite sum after computing full Fourier Series

I have computed the Full Fourier Series of the function $\phi:[-\pi,\pi] \rightarrow \Bbb{R}$ defined by $\forall x \epsilon[-\pi,\pi], \phi(x)=|\sin(x)|$ to be: $$ \phi(x) = {2\over\pi}+{1\over\pi} ...
3
votes
6answers
128 views

Proving $\big(n!^{\frac1n}\big)_{n\in\mathbb N^*} \to \infty$ [duplicate]

By definition, for any $a\in\mathbb R$, there exists $k\in\mathbb N^*$ such that, if $n\in\mathbb N+k$, then $n!^{\large\frac1n}>a$. Therefore, by induction, I must: find some $k$ that satisfies ...
3
votes
1answer
61 views

A Fibonacci series

Let $F_n$ be the $n^{th}$ term of the Fibonacci sequence. That is, $F_1 = F_2 = 1$ and $F_n$ is defined recursively for $n\geq3$ by $F_n = F_{n-2}+F_{n-1}$. It is a known fact that $$ ...
0
votes
1answer
39 views

Proving Banach's fixed point theorem

The hint tells me how to proceed but I am stuck. I define the sequence ${z_n}$ as is stated in the hint, First off, I want to prove that $|z_{n+k} - z_n| < \epsilon$ I add $z_{n+k-1}$ and ...
7
votes
4answers
100 views

Convergence of a sequence (possibly Riemann sum)

Let $a_1, a_2, a_3, . . . , a_n$ be the sequence defined by $$ a_n = 2\sqrt{n}-\sum_{k=1}^{n}\frac{1}{\sqrt{k}} = 2\sqrt{n} - \frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}-...-\frac{1}{\sqrt{n}} $$ show ...
-1
votes
1answer
49 views

Proof of a sequence with recursion

The problem asks to prove the following to be true. $$F^2_{n+1} - F_{n+1} F_n - F_n^2 = (-1)^n$$ Anyway, I've tried looking at this or similar proofs for going on an hour now, pretty much the only ...
1
vote
2answers
28 views

Non-Increasing sequence

Prove that the sequence ($a_n$) is eventually non-increasing for $ n\ge1$ $$a_n=\frac{3n+4}{2n-3}$$ Is it ok to say that for $a_n$ to be eventually non-increasing $a_{n+1} \le a_n$. Then work out the ...
0
votes
0answers
31 views

Show that $f_n(\cdot):[0,1]\rightarrow \Bbb R$, $f_n(x)=x^n(1-x^n)$ is simple convergent, but not uniform convergent.

Not sure if I solved this correctly. Show that $$f_n(\cdot):[0,1]\rightarrow \Bbb R$$ $$f_n(x)=x^n(1-x^n)$$ is simple convergent, but not uniform convergent. How I solved it: $(1)$ $\forall x \in ...
0
votes
2answers
53 views

Use the squeezing theorem to find the limit of the sequence

Is anyone able to help me answer this question? Or point me in the right direction? Use the squeezing theorem to find the limit of the sequence $\{a_n\}_{n=1}^{\infty}$ with $n$-th term ...
1
vote
2answers
54 views

l'Hopitals rule - is my working correct?

Is anyone able to help me with this question on l'Hopital's rule? Use l'Hopital's rule to find the limit of the sequence $\{a_n\}_{n=1}^\infty$ with $n$-th term $\displaystyle a_n = ...
0
votes
0answers
46 views

When the series $\sum\limits_{n=1}^{\infty} (\sqrt{n+a}-\sqrt[4]{n^2+n+b})$ converge and diverge

How to, depending on the real parameters a and b, determine when will the series converge and when will it diverge: $\sum\limits_{n=1}^{\infty} (\sqrt{n+a}-\sqrt[4]{n^2+n+b})$. I got this for homework ...
5
votes
4answers
267 views

Convergent or divergent

For homework (Calculus 2) I have to determine does this series converge or diverge and I don't know how to start: $$\sum\limits_{n=1}^{\infty} \dfrac {\ln(1+e^{-n})}{n}. $$
2
votes
5answers
96 views

How to prove/show that the sequence $a_n=\frac{1}{\sqrt{n^2+1}+n}$ is decreasing?

How to prove/show that the sequence $a_n=\frac{1}{\sqrt{n^2+1}+n}$ is decreasing? My idea: $n^2<(n+1)^2 /+1$ $n^2+1<(n+1)^2+1/ \sqrt{}$ $\sqrt{n^2+1}<\sqrt{(n+1)^2+1}/+n$ ...