-2
votes
1answer
28 views
2
votes
3answers
45 views

Fibonacci Sequence, Golden Ratio

I've been asked to show that $x_n \rightarrow L$ as $n \rightarrow \infty$ where $x_n = F_{n+1}/F_{n}$ for $n \in \mathbb{Z}^+$, where $F_n$ denotes the $n^{th}$ Fibonacci number. I am supposed to use ...
0
votes
1answer
23 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
59 views

Very complicated limit and trying to find convergence

I have no idea how to prove this: $\lim_{n \to \infty} \frac{2^2 \times4^2\times6^2\times\dots\times(2n)^2}{(1\times3)(3\times5)\dots((2n-1)(2n+1))} = \lim_{n \to \infty}\frac{2^2 ...
0
votes
2answers
49 views

Proof: $ M=\{x\in \Bbb{R}|\exists n \in \Bbb{N},\forall m \in \Bbb{N}( m \geq n\to a(m)\leq x)\}$ is bounded below

I need the proof of the following: Prop.: let be $a \in \Bbb{R}^\Bbb{N}$, and $a$ is bounded above, then: $$ M=\{x\in \Bbb{R}|\exists n \in \Bbb{N},\forall m \in \Bbb{N}( m \geq n \to a(m)\leq x)\} ...
7
votes
3answers
428 views

How to find the sum of $i(i+1)\cdots(i+k)$ for fixed $k$ between $i = 1$ and $n$?

I learned that $$\sum \limits_{i=1}^n i(i+1) = \frac{n(n+1)(n+2)}{3}$$ or in general $$\sum \limits_{i = 1}^n i(i+1)(i+2) \dots (i + k) = \frac{n(n+1)\dots (n+k+1)}{k+2}$$ From a mathematical ...
11
votes
2answers
660 views

Why is this allowed? (“Fourier's Trick”; finding the coefficients in a Fourier Series)

In my textbook (Introduction to Electrodynamics, D. Griffiths), we derive the equation for some strange potential function. Eventually, we get to this (for $n \in \mathbb{Z}^+$): $$ V_0(y) = ...
1
vote
2answers
65 views

Why does $\lim_{n \to\infty}a_{n+1} = \lim_{n\to\infty}a_n$?

Assume that $\{a_n\}$ is a convergent sequence. How to use the definition of a limit of a sequence to prove that
1
vote
2answers
90 views

Proof: Subsequence of n integers is divisible by n?

So I'm confused and stuck on how to approach this question. Any direction in the right path would be greatly appreciated. Let $n\in N$. Prove that any sequence of $n$ integers $a_1, a_2, ... a_n$ (no ...
1
vote
2answers
75 views

Use epsilon-N definition of a convergent sequence to prove that the series diverges

I'm stuck on this question - Use the ε-δ definition of a convergent sequence to prove that a series is divergent. I've attached the exact question but I have no idea what to do especially since it's ...
5
votes
1answer
223 views

Tricks. If $\{x_n\}$ converges, then Cesaro Mean converges (S.A. pp 50 2.3.11)

Show if $\{x_n\}$ is a convergent sequence, then the sequences given by the averages $\{\dfrac{x_1 + x_2 + ... + x_n}{n}\}$ converges to the same limit. (Not a duplicate) Let $\epsilon>0$ be ...
0
votes
1answer
30 views

$\exists L \in \Bbb{R}(\forall n \in \Bbb{N}(a^\frac{\lfloor b \cdot 10^n\rfloor}{10^n} \leq L))$?

Let be $a \in \Bbb{R}^{>0}, b \in \Bbb{R}$, $\exists L \in \Bbb{R}(\forall n \in \Bbb{N}(a^\frac{\lfloor b \cdot 10^n\rfloor}{10^n}\leq L ))$? I thought: if $a=1 \to 1^\frac{\lfloor b \cdot ...
0
votes
2answers
56 views

Proof of $\lim_{n\to\infty}\left(\frac{a_n+b_n}{2}\right)^n=\sqrt{ab}$

Let $a_n$ and $b_n$ two strictly positive sequences such that $$\lim_{n\to\infty}a_n^n=a>0\qquad \lim_{n\to\infty}b_n^n=b>0.$$ I need to prove that ...
1
vote
0answers
47 views

How to prove: If $a \to -\infty $ and $b$ is bounded from below by a constant $k\in\Bbb R^{>0}$, then the $a\cdot b\to -\infty$

I must proof the following, with $a: \Bbb{N} \to \Bbb{R}$ and $b: \Bbb{N} \to \Bbb{R}$ If $a \to -\infty\ (n\to\infty)$ and $b$ is bounded from below by a constant $k\in\mathbb R^{>0}$, then the ...
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, $$\prod_{n=1}^\infty \bigg(1-\dfrac{x^n}{x_{n+1}}\bigg)=e^{-x}$$ ...
0
votes
0answers
59 views

Using the root test twice.

Lets for example consider the following seires $\sum (something)^{n^2}$ Can I use the root test twice here and get $(something)$ then calculate the limite and then decide if the series is convergente ...
3
votes
3answers
103 views

Question on Rudin sequences?

In baby Rudin, Rudin shows that $$\lim_{n \to \infty}\sqrt[n]{p} = 1.$$ In the proof of limit he tries to prove that the limit is $1$. So he takes $x_n = \sqrt[n]{p} - 1$. I have never noticed this ...
0
votes
0answers
30 views

Practical methods to prove uniform convergence?

Can you please suggest some practical methods to prove whether a series of function (resp a sequence) is point-wise or uniformly convergent?
2
votes
1answer
38 views

Proof regarding series

I've encountered a recommended practice proof that I'd like some assistance in starting. Suppose that $\sum_{i=1}^\infty an$ and $\sum_{i=1}^\infty bn$ are both series with all positive terms and ...
1
vote
1answer
49 views

Proof of divergence of a series

I'd really appreciate some help with this question on my recent math assignment: Show that if $a_n > 0$ and $\lim_{n\to \infty} na_n = L$, where $0 < L < \infty$, then $a_n$ is divergent. ...
0
votes
3answers
61 views

Cauchy convergent sequences

Suppose that $(a_n)$ and $(b_n)$ are convergent sequences and that $b_n > 0$ for all $n$. Is it true that $(a_n / b_n)$ is Cauchy? If it is true, prove it. If it is not true, give a counterexample ...
0
votes
2answers
57 views

Sequence and convergence of subsequences

Suppose that $(a_n)$ is a sequence. Assume that both $(a_{2n})$ and $(a_{2n+1})$ converge to the same $L$. Prove carefully that $(a_n)$ also converges to $L$ I was thinking that $(a_{2n})$ and ...
2
votes
1answer
171 views

How do I use telescopic cancellation to prove the Fibonacci Sum Identity

I am reading a textbook which attempts to prove the Fibonacci Sum Identity by rearranging the Fibonacci recurrence relation as follows and then using telescoping cancellation to prove the identity: ...
1
vote
1answer
51 views

Proving the form of a sequence's terms

How do I go about attacking this problem and what is it asking? Suppose that $\alpha^2 = \alpha + 1$ and suppose $F_n$ denotes the Fibonacci sequence. Show that $\alpha^3 = 2\alpha +1, \alpha^4 = ...
1
vote
1answer
33 views

Recursive sine sequence and non recursive expression

Let $(a_n)$ be the sequence defined as $$a_{n} = \begin{cases}\sin(a_{n-1})&\text{if $n>0$}\\k &\text{if $n=0$}\end{cases}$$ where $k\in[0,2\pi)$. Prove that ...
2
votes
3answers
47 views

Convergence of recursive sequence convergence iff

Let $\{b_n\}_n \subseteq \mathbb{R}^+$. If the sequence $\{a_n\}_n$ is defined as $$a_n = \begin{cases}a_{n-1}+\frac{b_{n-1}}{a_{n-1}}&\text{if n>1}\\k&\text{if n=0}\end{cases}$$ where ...
0
votes
2answers
199 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? ...
1
vote
1answer
34 views

Proving that $\frac{1}{2} \le \frac{1}{2^n+1} + \frac{1}{2^n+2} + … + \frac{1}{2^n + 2^n}$

I'm having trouble proving that: $$\frac{1}{2} \le \frac{1}{2^n+1} + \frac{1}{2^n+2} + ... + \frac{1}{2^n + 2^n}$$ Edit: The next step is actually a mistake. I've put up a comment to the accepted ...
0
votes
1answer
48 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. ...
1
vote
1answer
55 views

Fixing $\epsilon$ in Proof that Any Convergent Sequence is Bounded

I am trying to understand the proof of the proposition: Any convergent sequence is bounded. In my textbook, the author uses the definition of convergence for a sequence $\{a_n\}\to l$ and fixes ...
7
votes
1answer
85 views

Prove $\frac {1}{\cos 0^\circ \cdot \cos 1^\circ} + \ldots +\frac {1}{\cos 88^\circ \cdot \cos 89^\circ}= \frac{\cos 1^\circ}{\sin 1^\circ}$

Prove the following identity: $$\frac {1}{\cos 0^{\circ} \cdot \cos 1^{\circ}} + \ldots +\frac {1}{\cos 88^{\circ} \cdot \cos 89^{\circ}} = \frac{\cos 1^{\circ}}{\sin 1^{\circ}}$$ After hours of ...
1
vote
2answers
266 views

Finding a non-recursive formula for a recursively defined sequence

So I have a recursive definition for a sequence, which goes as follows: $$s_0 = 1$$ $$s_1 = 2$$ $$s_n = 2s_{n-1} - s_{n-2} + 1$$ and I have to prove the following proposition: The $n$th term of the ...
2
votes
3answers
290 views

Monotone and Bounded Sequences: Proof

Say $s_1=1$ and $s_{n+1}=\frac{1}{5}(s_n+7)$ for $n\geq 1$. Prove that the sequence is monotone and bounded, then find the limit.
1
vote
3answers
84 views

Proving that a sequence is between certain values at certain n

I'm given that $a_1=1$, and for every $n \gt1, a_{n+1} = a_n + \frac{1}{a_{n}}$. I need to prove that $20 < a_{200} < 24$. I tried finding a limit at infinity setting both limits to $L$ ( for ...
1
vote
2answers
845 views

Accumulation Points & Convergence: A Sequence Existence Proof

Request: Prove that $x$ is an accumulation point of a set $S$ iff there exists a sequence $(s_n)$ of points in $S\setminus \{x\}$ such that $(s_n)$ converges to $x$. Attempt: Since this is a ...
3
votes
1answer
87 views

If $(s_n)$ converges to $s$, then $(\lvert s_n\rvert)$ converges to $\lvert s\rvert$.

Question: If $(s_n)$ converges to $s$, then $(\lvert s_n\rvert)$ converges to $\lvert s\rvert$. Prove or give a counterexample. Attempt: The statement is true because if $(\lvert ...
2
votes
2answers
318 views

On the Divergence of $s_n=\cos{\frac{\pi}{3}n}$: A Proof

Question: Show that $s_n=\cos{\frac{\pi}{3}n}$ is divergent. Attempt: Suppose that $\lim_{n\rightarrow \infty}(\cos{\frac{\pi}{3}n})=s$, then given an $\epsilon$, say $\epsilon=1$, we can ...
1
vote
1answer
54 views

Prove that $a_n \times b_n \to 0$ for $n \to \infty$

I want to prove this example: If $a_n \to 0$ for $n \to \infty$ and $(b_n)_n$ is bounded. Prove that $a_n \times b_n \to 0$ for $n \to \infty$. My first guess is that I should use the definition ...
5
votes
2answers
613 views

Proving the AM:GM inequality

I am doing past exam papers preparing for the finals and I came across this questions about three times: Prove that: $$\frac{a_{1}+a_{2}+\cdots+a_{n}}{n}\geq \sqrt[n]{a_{1}.a_{2}...a_{n}}$$ ...
5
votes
2answers
583 views

Nested Radical of Ramanujan

I think I have sort of a proof of the following nested radical expression due to Ramanujan for $x\ge 0$. $$\large x+1=\sqrt{1+x\sqrt{1+(x+1)\sqrt{1+(x+2)\sqrt{1+\cdots}}}}$$ for $ x\ge -1$ I ...
0
votes
1answer
403 views

Proving that the mothersequence converges to $x$ if any subsequence contains a subsequence which converges to $x$

Dear reader of this post, I am currently working on some problems about sequences and their subsequences. I proved a claim and because this prove involves some elementary concepts, I would like to ...
3
votes
1answer
72 views

How to Find the Radius of Convergence for This Proof?

Prove that if $\sum_{k=0}^{\infty}a_k$ converges, $\sum_{k=0}^{\infty}{a_k}{x^k}$ converges uniformly on $[0, 1]$. I posted this question a few days ago and was given a clue. I think that I'm ...
0
votes
2answers
31 views

Solving for $n$ in a geomtric progression

Given the general term of geometric sequence: $a_n = \dfrac{x}{2^n}$ I would like to solve for the value of n that makes $a_n =1$. My work so far: \begin{align*} a_n &= \frac{x}{2^n}\\ 2^n &= ...
0
votes
0answers
69 views

Prove limit using $\epsilon-M$ prove, To show $\lim_{n->\infty} \frac{n^2+2n}{n^3-5}$

To show $\lim_{n->\infty} \frac{n^2+2n}{n^3-5}$so here is my approach, Im not sure about something, just work out the preproof here, So I will first guess limit is 0, then start my preproof ...
3
votes
1answer
165 views

Need help to prove

I got the result below during my research. $$1=\frac{1}{1+a_1}+\frac{a_1}{(1+a_1)(1+a_2)}+\frac{a_1a_2}{(1+a_1)(1+a_2)(1+a_3)}+\frac{a_1a_2a_3}{(1+a_1)(1+a_2)(1+a_3)(1+a_4)}+... \tag 1$$ ...
3
votes
1answer
103 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
3answers
56 views

Prove that $\frac{1}{\sqrt{2n-1}}-\frac{1}{2n}\geq \frac{1}{2n}$ for $n = 1, 2, 3,…$

Prove that $\frac{1}{\sqrt{2n-1}}-\frac{1}{2n}\geq \frac{1}{2n}$ for $n = 1, 2, 3,...$ This is required to prove that the series $1-\frac{1}{2}+\frac{1}{\sqrt{3}}-\frac{1}{4}$ is divergent, but I ...
3
votes
1answer
226 views

Finding a Linear Recurrence Relation

A model for the number of lobsters caught per year is based on the assumption that the number of lobsters caught in a year is the average of the number caught in the two previous years. ...
6
votes
0answers
282 views

Extension of the Jacobi triple product identity

The Jacobi triple product identity is: $$\prod\limits_{n=1}^{ \infty }(1-q^{2n})(1+zq^{2n-1})(1+z^{-1}q^{2n-1})=\sum\limits_{n = - \infty }^ \infty z^n q^{n^2} $$ I would like to extend the idea for ...
1
vote
0answers
146 views

Power Series and Matrices

I am trying to prove that if a function $f(x)$ can be written as a power series in the form \begin{equation} f(x)=\sum_{n=0}^{\infty}c_n(x-x_0)^n \end{equation} such that $|x-x_0|<r$, then ...