0
votes
2answers
60 views

Proving Alternating Series Convergence

Suppose $x_n > 0$ and $\sum_{n=0}^\infty x_n$ is convergent. Prove that $\sum_{n=0}^\infty (-1)^nx_n$ is convergent. Any hints or starting points? So far I figured that I should show that the ...
1
vote
1answer
25 views

Proving Monotonic Sequence Theorem

A sequence $b_n$ is decreasing and bounded. Prove it it convergent. Proof: Since $b_n$ is bounded, $b_n > L$ where L is the greatest lower bound as per the completeness Axiom. Consider some ...
3
votes
1answer
53 views

Do I need to use induction to have sufficient rigor in this proof?

I'm taking my first analysis class this summer. The professor asked us to prove that $a^{2n}-b^{2n}$ is divisible by $a+b$. After dorking around with the first couple of $n$ I was able to come up ...
4
votes
3answers
98 views

Existence of increasing sequence $\{x_n\}\subset S$ with $\lim_{n\to\infty}x_n=\sup S$

Let $S\subset\mathbb{R}$ be a non-empty bounded above set. Then there exists a monotone increasing sequence $\{x_n\}\subset S$ such that $$\lim_{n\to\infty}x_n=\sup S.$$ I'm struggling with ...
0
votes
2answers
37 views

I know the limit of a subsequence exists, but how do I find it?

I know intuitively that for: $(a_n) = (1,1,2,1,2,3,1,2,3,4, ...)$ for any $m$ in the positive natural number there is a subsequence (m,m,m,..) that tends to m as n tends to infinity. I believe I ...
1
vote
2answers
57 views

Help with a proof I can't quite

Let $(F_j)^\infty_{j=1}$ be the sequence of Fibonacci numbers. For all $n \in \mathbb{N}$, $\sum\limits_{k=1}^{2n-1}F_kF_{k+1}=(F_{2n})^2$. I handled the base case quite well but couldn't go very ...
-3
votes
1answer
36 views

Proof of Theorem of Divergent Sequences [closed]

Let's say $(a_n)$ and $(b_n)$ are divergent sequences. Show or disprove the following: $((a_n b_n)_n)$ is divergent. $((a_n + b_n)_n)$ is divergent. $((c b_n)_n)$ with $c\neq0$ is divergent. ...
3
votes
2answers
50 views

Proving limit of a sum

I need to prove the following: $$\lim_{s \to \infty} \sum_{n = 1}^\infty \frac 1 {n^s} = 1$$ This is my attempt: \begin{align} \lim_{s \to \infty} \sum_{n = 1}^\infty \frac 1 {n^s} & = \lim_{s ...
-2
votes
1answer
29 views
0
votes
1answer
25 views

Need help finding a formula for this sequence

A sequence $(x_j)^\infty_{j=0}$ satisfies $x_1=1$, and for all $m \ge n \ge 0 $ $x_{m+n}+x_{m-n} = \frac12 (x_{2m}+x_{2n})$. I have to find a formula for $x_j$ and then I can prove that later for ...
0
votes
1answer
74 views

Proving a Sequence Diverges

Let {${\alpha_n}$} be a sequence in $\mathbb R$ satisfying $|\alpha_n - \alpha_{n+1}|\geq c$ for some $c > 0$ and all $n \in \mathbb N$. Prove that the sequence {$\alpha_n$} diverges. I've looked ...
-2
votes
2answers
42 views

Prove that $a_n$ = $2^n$ + $(-1)^n$

I am given the sequence $a_0= 2$, $a_1= 1$ and $a_{n+2}= a_{n+1} + 2a_n$ How do I prove that $a_n$ = $2^n + (-1)^n$
2
votes
0answers
100 views

Absolute Convergent Series Laws (Exercise)

Me again, I have troubles to understand the concept of absolute convergence in the general setting when the set can be uncountable. In the book what I read says: Definition: Let $X$ be a set ...
1
vote
1answer
60 views

Let $x$ be a real number. To prove…

Let $x$ be a real number. Define the sequence $(x_n)_{n\ge1}$ recursively by $x_1=1$ and $x_{n+1}=x^n+nx_n$ for $n\ge1$. Prove that, $$\prod_{n=1}^\infty \bigg(1-\dfrac{x^n}{x_{n+1}}\bigg)=e^{-x}$$ ...
3
votes
0answers
79 views

Interchange of finite sum with convergence sequences.

Hi everyone I'm wondering if the following proof is correct (to be honest at the beginning I have some troubles to understand what the sequences of the sums of convergent sequences has to be, but I ...
1
vote
2answers
37 views

Reasoning why the implication $t - \epsilon \le x \le t + \epsilon$ for $\epsilon \ge 0 \Rightarrow x = t$ holds using sequences.

In texts I've seen the following reasoning used several times: Suppose $t - \epsilon \le x \le t + \epsilon$ holds for $\epsilon \ge 0$. Then it in particular holds for $t - \frac 1 n \le x \le t + ...
0
votes
5answers
102 views

Proof of the limit of a sequence

The sequence is: $a_n$ = $\frac 1n$ [$(\frac 1n)^2 + (\frac 2n)^2 + (\frac 3n)^2...(\frac nn)^2$] The objective is a proof of the limit from 1 to infinity. Just from toying around with a few ...
0
votes
2answers
398 views

Proving a Sequence Does Not Converge

I have a sequence as such: $$\left( \frac{1+(-1)^k}{2}\right)_{k \in \mathbb{N}}$$ Obviously it doesn't converge, because it alternates between $0,1$ for all $k$. But how do I prove this fact? ...
2
votes
1answer
42 views

The limit of a sequence when at $n-1$

Suppose $\sum\limits_{n=1}^{\infty} a_n$ is a series that converges. Therefore, $\lim\limits_{n \to \infty} S_n$ exists, where $S_n$ is the sum of the first $n$ terms of the series. So, let ...
0
votes
2answers
273 views

Proof that changing a finite number of terms in a series does not change where or not it converges

I want to prove the following theorem: Changing a finite number of terms in a series does not change whether or not it converges, although it may change the value of its sum if it does converge ...
0
votes
1answer
58 views

Convergent subsequence

1) Let (x_n) be a sequence and let L ∈ R. Suppose that for each ϵ > 0, {k ∈ N : x_k ∈ B(L; ϵ)} is infinite. Show that (x_n) has a subsequence converging to L. ...
3
votes
2answers
276 views

Prove $\frac{1}{2} + \cos(x) + \cos(2x) + \dots+ \cos(nx) = \frac{\sin(n+\frac{1}{2})x}{2\sin(\frac{1}{2}x)}$ for $x \neq 0, \pm 2\pi, \pm 4\pi,\dots$

I know that this can be proven inductively. However, I can't get passed the trig. I am pretty sure trig identities can show that the expression above is true for $n=0$, and that if the expression ...
0
votes
3answers
54 views

Convergence of sequence (write a proof)

I need to prove the following affirmation: If $ \lim x_{2n} = a $ and $ \lim x_{2n-1} = a $, prove that $\lim x_n = a $ (in $ \mathbb{R} $ ) It is a simple proof but I am having problems how to write ...
5
votes
1answer
121 views

showing $a_n = \frac{\tan(1)}{2^1} + \frac{\tan(2)}{2^2} + \dots + \frac{\tan(n)}{2^n}$ is not Cauchy

My gut telling me that the following sequence is not Cauchy, but I don't know how to show that. $$a_n = \frac{\tan(1)}{2^1} + \frac{\tan(2)}{2^2} + \dots + \frac{\tan(n)}{2^n}$$
0
votes
2answers
89 views

Prove $\left(\frac{1}{n}+\frac{(-1)^n}{n^2}\right)$ converges to $0$ as $n\to\infty$

Using the formal definition of convergence of a sequence, show that the sequence converges to 0 as n tends to infinity. So we want to show that for every $\epsilon>0$, there exists $N$ such that ...
2
votes
2answers
326 views

Nested roots sequence, how to prove it's monotone and bounded?

Let $a\ge1$ and define the sequence $(x_n)$ recursively by: $$x_1 = \sqrt{a}$$ $$x_{n+1}= \sqrt{a+x_n}$$ Here's what I did: Plugging in some values makes it seem as if the sequence is increasing. I ...
0
votes
5answers
91 views

Is this a valid proof to show $\frac{n^5}{2^n}$ diverges?

Claim: $a_n = \frac{n^5}{2^n}$ diverges Let M be arbitrary Then $$ \forall \; n \ge \text{max} \big\{ \big[ \sqrt[3]{M} \, \big] + 1 , 3 \big\} \\ n > \sqrt[3]{M} \\ \implies n^3 > M $$ And ...
4
votes
2answers
254 views

Completeness proof of $\ell^p$

Say $\{x_n\}$ is Cauchy in $\ell^p$ and $x$ is its pointwise limit. To argue that $x \in \ell^p$ would the following be correct: Let $\varepsilon > 0$ and let $N$ be s.t. $n,m > N$ ...
0
votes
3answers
62 views

Limit of $a_n = -\cos \left( \frac{\pi}{8n-2} \right)$ (proof)

Warning to anyone who stumbles upon this: It is wrong completely and utterly don't use it for reference, thank you Don and Gerry for helping me see this So my first question is asking whether or not ...
2
votes
1answer
649 views

lim sup inequality proof - is this the right way to think?

I have tried to read many proofs of this but I'm not sure I get it, so please bare with me. Show that $\lim_{n \rightarrow \infty} \sup (a_n+b_n) \leq \lim_{n \rightarrow \infty} \sup (a_n)+lim_{n ...
1
vote
1answer
155 views

prove absolute convergence $\sum_{n=0}^{\infty}\binom{x}{n}$

i am trying to prove that this series $\sum_{n=0}^{\infty}\binom{x}{n}$ is absolute convergent. let $\binom{x}{n}:=\frac{x^n}{n!}$ but i am stuck on the way, can someone please help me out. my steps ...
14
votes
7answers
887 views

$\infty - \infty = 0$ ?

I am given this sequence with square root. $a_n:=\sqrt{n+1000}-\sqrt{n}$. I have read that sequence converges to $0$, if $n \rightarrow \infty$. Then I said, well, it may be because $\sqrt{n}$ goes to ...
1
vote
2answers
172 views

Analyzing a sequence and continuity proof

I am trying to understand the following proof: Given $f$ is continuous, prove that for every convergent sequence $(x_n) \to a$ that $\lim_{k\to \infty}f(x_k) = f(a)$ So the prove goes like this ...
15
votes
3answers
637 views

prove $\sqrt{a_n b_n}$ and $\frac{1}{2}(a_n+b_n)$ have same limit

i am given this problem: let $a\ge0$,$b\ge0$, and the sequences $a_n$ and $b_n$ are defined in this way: $a_0:=a$, $b_0:=b$ and $a_{n+1}:= \sqrt{a_nb_n}$ and $b_{n+1}:=\frac{1}{2}(a_n+b_n)$ for all ...
3
votes
2answers
67 views

$a_{n}$ converges and $\frac{a_{n}}{n+1}$ too?

I have a sequence $a_{n}$ which converges to $a$, then I have another sequence which is based on $a_{n}$: $b_{n}:=\frac{a_{n}}{n+1}$, now I have to show that $b_{n}$ also converges to $a$. My steps: ...
2
votes
0answers
26 views

check for convergence $\frac{1}{2}(a_{n-1}+a_{n-2})$ [duplicate]

Possible Duplicate: Does $x_{n+2} = (x_{n+1} + x_{n})/2$ converge? i am asked to prove the convergence of this sequence. What given is, is this: $a_{0}:=a$, and $a_{1}:=b$ and for all ...
4
votes
4answers
175 views

Help in proving $ \left( 1 + \frac{1}{n} \right)^{n} \leq \sum\limits_{k=0}^{n} \frac{1}{k!} < 3 $.

I am trying to prove this statement for all $ n \geq 1 $ using induction: $$ \left( 1 + \frac{1}{n} \right)^{n} \leq \sum_{k=0}^{n} \frac{1}{k!} < 3. $$ I said: Base case $ n = 1 $: $$ \left( ...
2
votes
3answers
2k views

Prove that every bounded sequence in $\Bbb{R}$ has a convergent subsequence

I am to prove that every bounded sequence in $\Bbb{R}$ has a convergent subsequence. I am stuck not knowing how and where to start.
0
votes
3answers
1k views

Proof of sequence limit, using epsilon-delta method.

${\displaystyle (a_n)}$ is a sequence with ${\displaystyle a_n = \frac{1}{\sqrt{n}}}$. Proove that ${ \displaystyle \lim\limits_{n\to\infty}{a_n} = 0 }$, using epsilon-delta method. First of all, I ...
0
votes
2answers
1k views

How do you prove that a sequence diverges?

Specifically the sequence $\{(-2)^n\}$
0
votes
4answers
121 views

Proof writing: $t_n=s_{n+k}.$ prove $s_n \rightarrow s \iff t_n \rightarrow s$

First prove $s_n$ converges iff $t_n$ converges $t_n=s_{n+k}.$ prove $s_n \rightarrow s \iff t_n \rightarrow s$ This is obvious but I am not so sure how to write the proof. Have: $\forall ...
1
vote
1answer
142 views

Verifying this proof: $a_n$ is cauchy $\implies a^2_n$ is cauchy

Prove: $a_n$ is cauchy $\implies a^2_n$ is cauchy Proof: Let $a_n$ be a cauchy sequence. Then we know: $\forall_{\epsilon > 0} \exists N_1 s.t. \forall_{n, m} \implies |a_n - a_m| < \epsilon$ ...
0
votes
0answers
123 views

How to prove that $(1^3+2^3+\cdots+n^3)=(1+2+\cdots+n)^2$ [duplicate]

Possible Duplicate: Intuitive explanation for the identity $\sum\limits_{k=1}^n {k^3} = \left(\sum\limits_{k=1}^n k\right)^2$ How can one prove that ...
2
votes
3answers
168 views

Why does he need to first prove the sum of the $n$ integers and then proceed to prove the concept of arithmetical progression?

I'm reading What is Mathematics, on page 12 (arithmetic progression), he gives one example of mathematical induction while trying to prove the concept of arithmetic progression. There's something ...
21
votes
1answer
807 views

How to prove convergence of polynomials in $e$ (Euler's number)

These polynomials in $e$ converge to 2$$f(i)=e^i - i \sum_{k=1}^{i-1}\frac{(i-k)^{k-1}{e^{i-k}}{(-1)^{k+1}}}{k!}, \text{ where } i>1$$ This function goes to 2. I've calculated this with sage math ...
3
votes
3answers
290 views

Another way to go about proving Binet's Formula

As I showed in another question of mine, it is easy to prove that $$\tag{1}\phi^{n+1} =F_{n+1} \phi+F_{n }$$ given $F_1=1$ , $F_2=1$ , $F_{n+1}=F_n+F_{n-1}\text{ ; }n\geq2$. Now, extending $(1)$ ...
4
votes
2answers
1k views

Prove that a set consisting of a sequence and its limit point is closed

Can someone please check whether the following simple proof is "mathematical"? Is it correct, complete, rigid? Can it be simplified? I'm a complete autodidact so I'm looking for someone to give me ...
4
votes
1answer
455 views

limit superior of a sequence proof

Let $(x_{n})\in\mathbb{R}^{+}$ be bounded and let $x_{0}=\lim\sup_{n\rightarrow\infty}x_{n}$. $\forall\epsilon>0$, prove that there are infinitely many elements less than $x_{0}+\epsilon$ and ...
3
votes
1answer
704 views

Justifying exchange of limits in a double sum - a dubious proof

It is a standard theorem (given in Rudin's Principles of Mathematical Analysis, page 175, and many other places), that if $\{a_{ij}\}$ is a doubly indexed sequence, and $$\sum_{j=1}^\infty |a_{ij}| = ...
1
vote
2answers
70 views

A lemma of convergence

While I tried to show, that any differentiable function is also steady, I found this lemma but wasn't able to show it. Is it true? Given two sequences $a_n$ and $b_n$, such that ...