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

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0
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2answers
27 views

Prove that a sequence converges to an element $x_0$ in $(X, d_1)$ if and only if the sequence converges to $x_0$ in $(X, d_2)$

Let $X$ and two strongly equivalent metrics $d_1, d_2 : X \times X \to \mathbb R$ on $X$ be given. Prove that a sequence $x_n$ converges to an element $x_0$ of $X$ in $(X, d_1)$ if and only if the ...
0
votes
1answer
494 views

Cantor's proof that every bounded monotone sequence of real numbers converges

Cantor constructed the field of real numbers by using Cauchy sequences. According to him every Cauchy sequence of real numbers converges (correct me if I'm wrong). So how did he prove that every ...
2
votes
1answer
35 views

Show $f(x) = \sum_{k = 0}^\infty a_kx^k$ has a positive radius of convergence.

Suppose that $ (a_k)_{k = 0}^\infty$ is a bounded sequence of real numbers. Prove that $f(x) = \sum_{k = 0}^\infty a_kx^k$ has a positive radius of convergence. Attempt: Suppose $\sum_{k = ...
1
vote
2answers
47 views

Fourier series question

I am just a beginner in Fourier series.How should I get start to tackle this question and show the partial sum has extrema? I have no clue to this question. Any help would be highly appreciated.
2
votes
2answers
88 views

Finding power series solution to differential equation $(1-x)y'=y$ centered at $ x=0$

I'm trying to teach myself how to solve differential equations with power series. I am stuck on working through $$(1-x)y'=y \text{ centered at } x= 0.$$ I've gotten to the point where I must figure ...
1
vote
0answers
63 views

For which values of $r$ the sequence $nr^n$ converges?

We have the sequence $$nr^n$$ and we want to find the values of $r$ such that $nr^n$ converges. What I did: for $|r|<1$ we have $r = \frac{1}{k}$ where $k>1$ or $k<-1$ so: ...
2
votes
2answers
57 views

Prove if $a_n$ converges to $0$ and $b_n$ is bounded, then $a_n b_n$ converges to $0$

We have these two hypotesis: $$\forall\epsilon_1>0, \exists n_0 | n>n_o \implies |a_n|<\epsilon_1$$ $$|b_n|<M$$ where $M$ is the sequence bound. Therefore, I've used hyp 2 to multiply ...
19
votes
0answers
665 views

About the integral $\int_{0}^{+\infty}\sin(x\,\log x)\,dx$

It is an interesting exercise to show that the function $f(x)=\sin(x\log x)$ is Riemann-integrable over $\mathbb{R}^+$ (as shown by robjohn in this related question, for instance). Even more ...
0
votes
3answers
50 views

Show that for large $n$, $(1-\frac{\lambda}{n})^n$ is approximately $exp(-\lambda)$

In their book on Probability, Grinstead a Snell say (page 189) that for large $n$, $(1-\frac{\lambda}{n})^n $ is approximately equal to $exp(-\lambda)$ Using the binomial formula I can expand ...
0
votes
1answer
44 views

Find where this series uniformly converges [closed]

Given the following series: $$f_n(x) = \frac{x^2}{(1+x^2)^n}$$ for $x\in\mathbb{R}$, and let $s_k = \sum_{n=0}^kf_n(x)$. Find values $a < b$ where the series uniformly converges on $[a, b]$. So ...
1
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0answers
33 views

Prove that $X_n=\frac{1}{n}$ is not a contractive sequence.

So I was trying to prove this using a contradiction. I started by stating the definition of contractive which is that there exist r in (0,1) such that for all n in N, |$X_{n+2}$-$X_{n+1}$|$\leq$ ...
1
vote
2answers
60 views

Uniformly converge with equicontinuous family

Let $\{f_n\}$ be a equicontinuous sequence and converge pointwise in a compact set $K$ of $\mathbb{R}^n$. Prove that the sequence converge uniformly in $K$. My attempt: Since $\{f_n\}$ is ...
1
vote
2answers
62 views

What is the limit of this sequence $x_0 = 1$, $x_1 = 2$, $x_{n + 2} = \frac{1}{2} \cdot (x_n + x_{n + 1})$

I have an exercise in my last assignment for calculus, where basically we have to find the limit of a sequence. We don't know the closed form, so it's quite complicated. We have maxima, but it does ...
0
votes
3answers
74 views

Proving $\sum_{n=0}^{\infty }\frac{1}{(48n+1)(48n+47)}=\frac{\pi}{2208}(\cot(\frac{\pi}{24})+\sec(\frac{11\pi}{24}))$

Proving $$\sum_{n=0}^{\infty }\frac{1}{(48n+1)(48n+47)}=\frac{\pi}{2208}(\cot(\frac{\pi}{24})+\sec(\frac{11\pi}{24}))$$ When I used the WolframAlpha, I got the the following result: I could ...
0
votes
2answers
23 views

Sequence. Finding the terms from given equations

The sum of three terms is $33$, and their product is $1287$. Find these terms. I tried to solve this, I went to a result but I'm not sure I was right.The given equations are: ...
0
votes
0answers
26 views

The series $n^{-1-1/n}$ diverges, but how do I show this? [duplicate]

I have problems with the following sum $n^{-1-1/n}$. I need to show that the sum diverges, I know that I have to use the comparison test after spending too much time with the wrong tests. My problem ...
1
vote
1answer
32 views

Find the value of the constants given the $n$th term of a sequence, the limit and the first two terms

The $n$th term of the sequence is $u_n$. The sequence is defined by $$u_{n+1}=pu_n+q$$ $p$ and $q$ are constants. The first two terms of the sequence are given by $$u_1=60$$ $$u_2=48$$ The limit of ...
1
vote
5answers
161 views

Why is $\lim(n!)^{1/n}=\infty$ [duplicate]

Why is $\lim\limits_{n\to\infty}(n!)^{1/n}=\infty$ It is more or less clear that the sequence is increasing by ratio test ...
0
votes
3answers
43 views

Find the remainder of the following

Given that $F(1)=1,F(2)=1$ and $F(n)=F(n-1)+F(n-2)$ , $n>2$ for this series, now if $f(n)$ is the remainder if $F(n)$ is divided by $5$ then the value of $f(2000)$ is 1 0 3 2 For doing ...
1
vote
3answers
55 views

Convergence series problem.

How to show that $$\sum_{n=3}^{\infty }\frac{1}{n(\ln n)(\ln \ln n)^p}$$ converges if and only if $p>1$ ? By integral test, $$\sum_{n=3}^{\infty }\frac{1}{n(\ln n)(\ln \ln n)^p}$$ ...
1
vote
2answers
61 views

Inequality: $\left|\sum_{i = n + 1}^{\infty} \frac{(-1)^i}{i + x}\right| \le \frac{1}{n + 1 + x}$

This (with $x > 0$ and $n \in \Bbb N$): $$ \left|\sum_{i = n + 1}^{\infty} \frac{(-1)^i}{i + x}\right| \le \frac{1}{n + 1 + x}$$ Was used to prove that the series of general term the summand on ...
3
votes
3answers
105 views

Residue of $\Gamma^{2}$ and $\Gamma^{3}$

Based on wiki, the residues of $\Gamma$ at non positive integers are given by: $$\text{Res}\left ( \Gamma(z),z=-n \right )=\frac{(-1)^{n}}{n!}.$$ I have been trying to find residue for $\Gamma^{2}$ ...
6
votes
1answer
50 views

Proving $ \liminf x_n=2 $

Let $ {x_n} $ be a real sequence, defined by, $x_n = \begin{cases} 2+\frac{1}{n}, & \text{if $n=m^{2}$ for some $ m\in \mathbb{N} $} \\ n+1, & \text{otherwise.} \end{cases}$ Here I need to ...
8
votes
2answers
1k views

An infinite product

I am trying to compute the infinite product $$ \prod\limits_{n=2}^\infty \left(1+\frac{1}{2^n-2}\right) . $$ Wolfram Alpha says the result is $2$, but I can't seem to figure out why.
0
votes
2answers
41 views

How to calculate the argument and its limit for the sequence $z_n=-2+i\frac{(-1)^n}{n^2}$

I am trying to show that the limit of the sequence $$z_n=-2+i\frac{(-1)^n}{n^2}$$ exists, using the polar representation. Note that $\lim_{n\rightarrow \infty }z_n=-2$. $$$$I am finding difficulty in ...
0
votes
1answer
51 views

Prove that ${f_n}$ converges uniformly to the zero function

Fix $a,b∈R$ with $a<b$. Define the sequence ${f_n}$ of functions by $f_n:[a,b]\to\mathbb{R}$ by $f_n(x)=x/n$. Prove that ${f_n}$ converges uniformly to the zero function. I somewhat understand how ...
0
votes
2answers
77 views

Decide on convergence of the series $\sum 1/r^{\ln n}$

I'm not sure how to go about this problem. The root and ratio tests are inconclusive and I'm not sure how to find another sequence for the limit comparison test. $$\sum_{n = ...
1
vote
2answers
62 views

Proof strategy for $\lim_{n\to\infty} \frac{2^n}{3^{n+1}}$

In order to take the limit $$\lim_{n\to\infty} \frac{2^n}{3^{n+1}}$$ I did: $$3^{n+1}>n2^n\implies \frac{1}{3^{n+1}}<\frac{1}{n2^n}\implies\\\frac{2^n}{3^{n+1}}<\frac{2^n}{n2^n}\implies ...
3
votes
3answers
135 views

How to show $\sum_{n=0}^{\infty }\frac{1}{(24n+5)(24n+19)}=\frac{\pi }{336}(\sqrt{6}-\sqrt{3}-\sqrt{2}+2)$

How to show $$\sum_{n=0}^{\infty }\frac{1}{(24n+5)(24n+19)}=\frac{\pi }{336}(\sqrt{6}-\sqrt{3}-\sqrt{2}+2)$$
2
votes
1answer
36 views

Proof that $\lim_{n\to\infty} a_n = L$ then $\lim_{n\to\infty} |a_n| = |L|$

In order to prove that $$\lim_{n\to\infty} a_n = L \implies \lim_{n\to\infty} |a_n| = |L|$$ I started with my assumption: $\lim_{n\to\infty} a_n = L$ we have: $$n>n_0\implies |a_n - L| < ...
0
votes
1answer
19 views

$U_{n+1} = a * U_n * (1-U_n)$ with : $3<a<3.6$ What are some values of $a$ such that $U_n$ changes its periodicity?

$U_{n+1} = a * U_n * (1-U_n) = a * U_n - a * (U_n)^2$ with : $3<a<3.6$ What are some values of $a$ such that $U_n$ changes its periodicity? I've computed $U_n$ for many different values of ...
1
vote
0answers
68 views

Prove that if $\lim_{n\to \infty} a_n=\infty$ then $\lim_{n\to \infty} \frac{1}{a_n}=0$

let $C= \frac{1}{\epsilon}$ There $\exists N\in\mathbb N$ such that for every $n>N$, it is true that: $$a_n>\frac{1}{\epsilon}$$ We should prove that for every $\epsilon>0$ there exists ...
1
vote
1answer
60 views

Using Taylor's Theorem and the Constancy Theorem, solve the following proof.

Using Taylor's Theorem and the Constancy Theorem prove that $\sqrt{1+x}=1+\frac{x}{2}+\sum_{n=2}^{\infty}(-1)^{n-1} \frac{1}{2n} \frac{(1- \frac{1}{2})(2- \frac{1}{2}) ... ((n-1)- ...
6
votes
1answer
172 views

Evaluate Integral with $e^{ut}\ \Gamma (u)^{2}$

I am trying to integrate this integral: $$f(x)=\frac{1}{2\pi j}\int_{c-j\infty}^{c+j\infty}x^{-s}\sigma ^{ms-m}\left [ \frac{\Gamma \left ( \frac{s}{\beta} \right )}{\Gamma \left ( \frac{1}{\beta} ...
5
votes
4answers
366 views

Easy Double Sums Question: $\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \frac{1}{(m+n)!}$

How to calculate $\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \dfrac{1}{(m+n)!} $ ? I don't know how to approach it . Please help :) P.S.I am new to Double Sums and am not able to find any good sources ...
7
votes
1answer
118 views

Showing $\sum_{n=1}^{\infty }(-1)^{n-1}\frac{(2n-2)!\zeta (2n)}{\pi ^{2n}}(1-\frac{1}{2^{2n}})(1+\frac{1}{2^{2n-1}})=\frac{\log(2)}{4}$.

How do I show that show $$\frac{0!\zeta (2)}{\pi ^2}\left(1-\frac{1}{2^2}\right)\left(1+\frac{1}{2}\right)-\frac{2!\zeta (4)}{\pi ...
-2
votes
3answers
85 views

Limit of $\frac{n^2}{4^n}$ [duplicate]

How can I prove that the limit of $n^2/4^n$ as $n$ approaches infinity is 0? I want to solve it without using de l'Hospital's rule and I tried some inequalities, but I don't find a nice solution.
3
votes
2answers
158 views

Quotient of Sequences

I have a question about convergent sequences which I have no idea how to go about. It seems intuitive to me that if we have two sequences $a_n$ and $b_n$ such that they both have a limit ($+\infty$ ...
2
votes
1answer
41 views

Writing an expression as a product of products

I am currently dealing with the following expression: $$\left(\prod_{i=1}^{N-1}(\lambda_N-\lambda_i)\right)\left(\prod_{i=1}^{N-2}(\lambda_{N-1}-\lambda_i)\right)\cdots (\lambda_2-\lambda_1)$$ Is ...
0
votes
1answer
35 views

Epsilon-N limit Proof for product law for limit

Let the $\lim_{n\to \infty} x_n=a$ and let $\lim_{n\to \infty} y_n=b$. Let $\epsilon>0$ for some $n\ge N$ My notes says: $|(x_n)(y_n)-ab|=|x_n(y_n-b)+b(x_n-a)|$ Can someone show me the ...
0
votes
1answer
28 views

Let $(s_n)$ and $(t_n)$ be bounded sequences of nonnegative numbers. Prove $\lim \sup{s_n,t_n}$ $\leq$ $(\lim \sup{s_n})(\lim \sup{t_n})$?

Let $(s_n)$ and $(t_n)$ be bounded sequences of nonnegative numbers. Prove $\lim \sup{s_n,t_n}$ $\leq$ $(\lim \sup{s_n})(\lim \sup{t_n})$? Can someone help explain this proof to me?
0
votes
1answer
256 views

Geometric Series Question Given Sum of First 2 and First 3 Terms

The sum of the first two terms of a convergent geometric series is 8 and the sum of the first three terms is 26. What is the sum of the series? I get to $ar^2 = 18$, $r = \sqrt{18/a}$ and then I ...
2
votes
2answers
52 views

Prove $\sum {b_n}^{1\over2}\dot\,{1\over{n^{\alpha}}}\,$ converges given $\sum b_n\,$ converges

If $b_n>0$ and $\sum b_n\,$ converges, prove $\sum {b_n}^{1\over2}\dot\,{1\over{n^{\alpha}}}\,$ converges for all $\alpha>{1\over2}$. I know ...
0
votes
1answer
32 views

Two sequence be convergent

I'm having trouble on figuring out how to show x_n and y_n are convergent. I get that the last part is Comparison Theorem. So if i can show the two sequences are convergent I can say it's the ...
0
votes
0answers
41 views

Can this be solved by Jensen's Inequality?

For a sequence of known constants $a_i$, $i=1, \dots, n$, I can define two values $$C_1 = \frac{1}{\bar{a}^2} = \frac{1}{\left(\frac{1}{n} \sum_{i=1}^n a_i \right)^2}$$ $$C_2 = ...
0
votes
0answers
27 views

Comparing two sequences that branch from one number

/// I'll be rewording or deleting this question over the next week - thanks. / There is a sequence of natural numbers produced by the two functions $f'(b) = 3b+1$ [where b is odd] and $f''(b) = ...
0
votes
1answer
37 views

Every Cauchy sequence is bounded. Question about a detail in the proof.

To prove that every Cauchy sequence is bounded, we say that after some $k$ all $x_n$ are contained in a ball of given radius $\epsilon$ for $n \geq k$. We then say all $x_n$ with $1 \leq n \leq k-1$ ...
0
votes
0answers
23 views

Uniform Convergence Complex Limit

I want to confirm my understanding for this question. ''If $\sum M_n$ is a convergent series of constants, and the real valued function $u_n(z) < M_n$ for all n, does $\sum u_n(z)$ converge ...
2
votes
4answers
57 views

Write down each of the terms in the expansion of $\sin(x^2)-(\sin(x))^2$.

Write down each of the terms in the expansion of $\sin(x^2)-(\sin(x))^2$. Taylor's Theorem applies at the point $a=0$ and with $n=4$. Got no idea how to proceed. My lecture notes have one example ...
0
votes
1answer
44 views

Sum of A Finite Series

I am trying to solve the sum of this finite series: $$\sum_{i=0}^n \frac{1}{6^n}.$$ I am having problems where to start, as it is completely different to the other ones I have done. Here is the ...