# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### Prove that for any integer 𝑛 ≥ 1, we have [closed]

Prove that for any integer 𝑛 ≥ 1, we have
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### 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 ...
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### 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 ...
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### 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 $a_n$ = $2^n + (-1)^n$
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### 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 ...
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### 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, $$\prod_{n=1}^\infty \bigg(1-\dfrac{x^n}{x_{n+1}}\bigg)=e^{-x}$$ ...
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### 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 ...
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### 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 ...
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### $\infty - \infty = 0$ ?

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 ...
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### 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 ...
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### 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 ...
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### $a_{n}$ 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: ...
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### 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 ...
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 $n = 1$: $$\left( ... 3answers 2k views ### Prove that every bounded sequence in \Bbb{R} has a convergent subsequence I am to prove that every bounded sequence in \Bbb{R} has a convergent subsequence. I am stuck not knowing how and where to start. 3answers 1k views ### Proof of sequence limit, using epsilon-delta method. {\displaystyle (a_n)} is a sequence with {\displaystyle a_n = \frac{1}{\sqrt{n}}}. Proove that { \displaystyle \lim\limits_{n\to\infty}{a_n} = 0 }, using epsilon-delta method. First of all, I ... 2answers 1k views ### How do you prove that a sequence diverges? Specifically the sequence \{(-2)^n\} 4answers 121 views ### Proof writing: t_n=s_{n+k}. prove s_n \rightarrow s \iff t_n \rightarrow s First prove s_n converges iff t_n converges t_n=s_{n+k}. prove s_n \rightarrow s \iff t_n \rightarrow s This is obvious but I am not so sure how to write the proof. Have: \forall ... 1answer 142 views ### Verifying this proof: a_n is cauchy \implies a^2_n is cauchy Prove: a_n is cauchy \implies a^2_n is cauchy Proof: Let a_n be a cauchy sequence. Then we know: \forall_{\epsilon > 0} \exists N_1 s.t. \forall_{n, m} \implies |a_n - a_m| < \epsilon ... 0answers 123 views ### How to prove that (1^3+2^3+\cdots+n^3)=(1+2+\cdots+n)^2 [duplicate] Possible Duplicate: Intuitive explanation for the identity \sum\limits_{k=1}^n {k^3} = \left(\sum\limits_{k=1}^n k\right)^2 How can one prove that ... 3answers 168 views ### Why does he need to first prove the sum of the n integers and then proceed to prove the concept of arithmetical progression? I'm reading What is Mathematics, on page 12 (arithmetic progression), he gives one example of mathematical induction while trying to prove the concept of arithmetic progression. There's something ... 1answer 807 views ### How to prove convergence of polynomials in e (Euler's number) These polynomials in e converge to 2$$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$$This function goes to 2. I've calculated this with sage math ... 3answers 290 views ### Another way to go about proving Binet's Formula As I showed in another question of mine, it is easy to prove that$$\tag{1}\phi^{n+1} =F_{n+1} \phi+F_{n }$$given F_1=1 , F_2=1 , F_{n+1}=F_n+F_{n-1}\text{ ; }n\geq2. Now, extending (1) ... 2answers 1k views ### Prove that a set consisting of a sequence and its limit point is closed Can someone please check whether the following simple proof is "mathematical"? Is it correct, complete, rigid? Can it be simplified? I'm a complete autodidact so I'm looking for someone to give me ... 1answer 455 views ### limit superior of a sequence proof Let (x_{n})\in\mathbb{R}^{+} be bounded and let x_{0}=\lim\sup_{n\rightarrow\infty}x_{n}. \forall\epsilon>0, prove that there are infinitely many elements less than x_{0}+\epsilon and ... 1answer 704 views ### Justifying exchange of limits in a double sum - a dubious proof It is a standard theorem (given in Rudin's Principles of Mathematical Analysis, page 175, and many other places), that if \{a_{ij}\} is a doubly indexed sequence, and$$\sum_{j=1}^\infty |a_{ij}| = ...
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 ...