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

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1answer
20 views

How to use the Maclaurin Series to get $f^{10}(0)$ of $f(x)=(\cos(3x^2)−1)/x^2$

I have no idea how to compute this. Any help is appreciated! Thank you. $f^{10}(0)$ of: $$ f(x)= \frac{\cos(3x^2)−1}{x^2} $$
-1
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2answers
36 views

How do I calculate the coeffient in an expansion? [closed]

Calculate coefficient in $x^3y^{21}$ in the expansion $(x-y)^{24}$.
1
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1answer
22 views

Series representation of a function. Generating the series formula.

in general say the question is to find the series representation of $ arctan(3x)$ the solution is the $\int \sum (-1)^n * (3x)^{2n}=$ $$ \sum (-1)^n * 3x^{2n+1}/(2n+1) $$ but my confusion is why ...
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1answer
29 views

Uniform convergence and differentiation proof of remark 7.17 in Rudin's mathematical analysis

Rudin page 152 Theorem 7.17: Suppose $\{f_n\}$ a sequence of functions, differentiable on $[a,b]$ and such that $\{f_n(x_0)\}$ converges for some point x0 on [a,b]. If {f′n} converges uniformly on ...
0
votes
1answer
80 views

Prove that a closed ball is closed

Prove that in $\[\vec{E}\]$ normed vector space $\[B(\vec{x}, \varepsilon )\]$ is a closed set. and $\[B'(\vec{x}, r)\]$ is an open set. Fo the first part I created a sequence ($\[x_{n}\] $) ...
2
votes
1answer
58 views

Show that $\sum_n \frac{1}{a_n}\lt90$ [duplicate]

Let $1,2,3,4,5,6,7,8,9,11,12,\cdots$ be the sequence of all the positive integers which do not contain the digit zero. Write $\{a_n\}$ for this sequence. By comparing with a geometric series, show ...
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0answers
36 views

Gauss' test for Convergence

In the text I am using, the hypotheses used for Gauss' test for convergence are different to others I have seen. The text has the hypotheses: if the series $\sum_{n=1}^{\infty} a_n$ is such that ...
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0answers
21 views

Sequence in $\mathbb Z^n$ gets dominated by sequence in $\mathbb Z$

Iam triying to proof with $| m |= (m_1 + \dots +m_n)^{1/2}$: Given a positive sequence $(d_m)_{m \in\mathbb Z^n}$ with $d_m \to 0$ as $|m|\to\infty$, there exists a positive sequence ...
4
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1answer
56 views

Different types of Convergence for a Series Function

I am currently investigating the convergence of the following function, $f(x)=\sum\limits_{k=1}^{\infty} \dfrac{x^{k}+\sin(k)}{k^{2}}$ for different "senses". I have shown that $f(x)$ converges ...
1
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1answer
36 views

Convergence of the series $\sum\limits_{n=1}^{\infty}\frac{a^{\log n}}n$

$$\sum_{n=1}^{\infty}\frac{a^{\log n}}n$$ The exercise requires to study the series for $a\in \mathbb{R} > 0$. The case $a\geq 1$ is very simple. I can't wrap my head around the remaining cases ...
3
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2answers
77 views

Let $u_{n+3} = u_n + 2u_{n+1}$ . Show that $p$ divides $u_p$ for all $p$ prime number.

Let $(u_n)$ a sequence such that $u_0 = 3$, $u_1 = 0$, $u_2 = 4$ and $u_{n+3} = u_n + 2u_{n+1}$ Show that $p$ divides $u_p$ for all $p$ prime number. I'm really stuck on this exercise, ...
0
votes
4answers
77 views

How is $f_{2000}$ equals $0$? [duplicate]

Let $(F_n)_{n\ge 1}$ be the sequence of numbers defined by $F_1=1=F_2$; $F_{n+1}=F_{n}+F_{n-1}$ for $n\geq2.$ Let $f_n$ be the remainder left when $F_n$ is divided by $5$. Then $f_{2000}$ equals ...
0
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1answer
34 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 ...
2
votes
3answers
174 views

Calculating $1+\frac13+\frac{1\cdot3}{3\cdot6}+\frac{1\cdot3\cdot5}{3\cdot6\cdot9}+\frac{1\cdot3\cdot5\cdot7}{3\cdot6\cdot9\cdot12}+\dots? $

How to find infinite sum How to find infinite sum $$1+\dfrac13+\dfrac{1\cdot3}{3\cdot6}+\dfrac{1\cdot3\cdot5}{3\cdot6\cdot9}+\dfrac{1\cdot3\cdot5\cdot7}{3\cdot6\cdot9\cdot12}+\dots? $$ I can see ...
0
votes
1answer
26 views

show that $(x_k)$ is convergent and limit$\notin\ell^p$

Let $\ell^p:=\{(x_n)|\|x_n\|_{\ell^p}<\infty\}$ be the space of sequences with finite $\ell^p$-norm. Show that the sequence $x_1:=1$, $x_k:=\frac{1}{\log k}$, $k\geq 2$ converges to $x=0$, but ...
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 ...
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2answers
20 views

A question about the divergence of a series.

The following is a criteria for divergence: $\exists N_0$ such that $\left|\frac{a_{n+1}}{a_n}\right|>1$ for $n\geq N_0$, then $\Sigma{a_n}$ is divergent. $1,-2,3,-4,5,-6,\dots$ satisfies ...
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0answers
27 views

Taylor Series Calc 2

I am not sure how to find a series representation for the natural log. If anyone can show me some helpful steps to solve this problem it would be greatly appreciated. What is the Maclaurin series ...
0
votes
1answer
44 views

Test whether the series $\sum_{n = 1}^\infty (n!)^n/(n)^{4n}$ diverges or converge.

I've found this series in my calculus's book $$\sum_{n = 1}^\infty \frac{(n!)^n}{n^{4n}}$$ But the correct form is $$ \sum_{n = 1}^\infty \frac{n^n}{n^{4n}} $$ My question is: How to verify if ...
2
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2answers
32 views

Two series sum, one converging, one diverging

I need to give an example of a two series sum a_n and b_n such that the lim a_n/b_n=1. One series has to diverge and one has to converge. a_n and/or b_n don't necessarily have to be positive. I have ...
2
votes
3answers
81 views

Why does $\prod_{n=1}^\infty \frac{n^3 + n^2 + n}{n^3 + 1}$ diverge?

$\prod_{n=1}^\infty \frac{n^3 + n^2 + n}{n^3 + 1}$ diverges, and I have no idea why? It would seem using L'hop, $\frac{n^3 + n^2 + n}{n^3 + 1}$ goes to 1. So it should end up just being ...
0
votes
2answers
45 views

Series - convergence and divergence

Does the following series diverge or converge? $$ \frac{1}{2} + \frac{1}{2^3} + \frac{1}{2\times3^2} + \frac{1}{4^3} + \frac{1}{2\times5^2} + \frac{1}{6^3} $$ I am not able to find a closed form ...
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3answers
35 views

Determine whether this series converges

I am studying for a calculus final and have come across this practice question: $\text{Determine whether the series is absolutely convergent:}$ $\sum\limits_{n=1}^{\infty} \frac{(-1)^n [1 \cdot 3 ...
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2answers
33 views

Connection between series

I have to show that if $\sum_{n=1}^{\infty} a_n$ is absolutely convergent then $\sum_{n=1}^{\infty} a_n^2$ is absolutely convergent too. Please give me some hint, how do I start the excercise. ...
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2answers
35 views

Do this series converge or diverge?

Let, $$\sum \limits _{n=1}^\infty \frac{(-1)^nn^2}{n^2+1} \space (1)$$ This is an alternate series, so I applied the Leibnitz test. Let $a_{n}=\frac{n^2}{n^2+1}$ be a sequence, if $a_{n}$ is ...
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3answers
27 views

how to prove a sequence is unbounded?

I'm little bit confused about how to prove that sequences are unbounded. For example, I have this sequence: $$A_1 = a,\quad (a>0, a \in \mathbb{R}),\\ A_{n+1} = A_n + (A_n)^2/(1+A_n)^2$$ I know ...
1
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3answers
37 views

Nth term of a sequence

First of, I know this is quite easy but I can't really work it out. I need find the rule for these sequences $a.$ $2, 3, 4.5, 6.25...$ $b$ $54, 18, 6, 2...$ $c$ $0.01, 0.1, 1, 10...$ My Steps ...
1
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1answer
44 views

$(a_n)_{n=1}^\infty$ is a convergent sequence and $a_n \in [0,1]$ for all $n$. Proof of limit $(a_n)_{n=1}^\infty$ lies in [0,1].

Textbook question: $(a_n)_{n=1}^\infty$ is a convergent sequence and $a_n \in [0,1]$ for all $n$. Proof of limit $(a_n)_{n=1}^\infty$ lies in [0,1]. I don't understand the question I suppose. It ...
1
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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 ...
2
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2answers
44 views

How to prove series convergence

I have this series: $$\sum \limits_{n=1}^\infty \left(\frac1n+\sqrt{1+n^2}-\sqrt{2+n^2}\right)^2$$ I know that it's convergent (from WolframAlpha) but I need to prove it is convergent. How can I do ...
0
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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 ...
1
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1answer
28 views

Determine the value of r where the series converges

show that $$ \big(r\big)^{ln(n)} = \big(n\big)^{ln(r)} $$ Then determine the values of r (with r>0) for which the series $$ \sum_1^\infty (\big(r\big)^{ln(n)})$$ converges. r must be in what ...
2
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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.
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1answer
36 views

If a sequence converges, then it is Cauchy?

Considering the following proof and its converse: If a sequence converges, then it is Cauchy. That is, if $\lim_{n\to \infty}a_{n} = L$, then given $m>N$, we have that $|a_{m}-a_{n}| < \epsilon$ ...
4
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2answers
34 views

Help with Convergence/Divergence

So I am trying to prove whether the following problem converges or diverges? $$\sum_{n=1}^\infty \left({n\over n+18}\right)^n$$ So I decided to use the Root test. $$ L = \lim_{n\to ...
0
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2answers
29 views

let $f_n(x) = \frac{x^n}{1 + x^n}$. determine whether $f_n \to f$ uniformly on $[0, 1]$ and whether $f_n \to f $ uniformly on $[0, \infty)$

this sequence was given as a practice problem and I'm really having trouble. Heres the question: let $f_n(x) = \frac{x^n}{1 + x^n}$. determine whether $f_n \to f$ uniformly on $[0, 1]$ and whether ...
0
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4answers
51 views

Why is the derivative of circle area the circumference? [duplicate]

Why is the derivative of the volume of a sphere the surface area? And why is the derivative of the area of a circle the circumference? Too much of a coincidence, there has to be a reason! Also, why ...
2
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1answer
28 views

Need help with Maclaurin Series for $\cos^2 x$

So I have been working on finding the Maclaurin series for $\cos^2 x$ I am thinking of using the identity: $\cos^2 x = {1\over 2}(1+\cos(2x))$ By using the known Maclaurin series for $cos(x)$, I ...
0
votes
2answers
32 views

Accumulation Points Questions.

Im studying accumulation points and the theory seems quiet simply however the questions i cant seem to figure out. $$(1+1/n)^n$$ I know the limit of this as $n\rightarrow\infty= e$, however is this ...
3
votes
2answers
24 views

Maclaurin Series for a natural logarithm

Can anyone please help me with this question? Find the Maclaurin series and the interval of convergence for $f(x) = \ln(1-7x^9)$ I thought the answer was $$\sum_{n=1}^{\infty} (-1)^n ...
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5answers
118 views

limit of $n^{\frac{1}{\sqrt{n}}}$

How can prove limit of $n^{\frac{1}{\sqrt{n}}}$ is 1 without using advanced theorem? I think we have to make sequence which is greater than $n^{\frac{1}{\sqrt{n}}}$ and its limit is also 1 with just ...
4
votes
2answers
44 views

Find all $x$ such that the series converges

Let $$\sum_{n=2}^{\infty}{\frac{\cos(nx)}{\ln(n)}}$$ I want to find all $x\in \mathbb R$ such that the series converges. I only know that the series converges for $x=\pi,\frac{\pi}{2}$ But I don't ...
2
votes
0answers
40 views

Show sum involving sines is non-negative

I want to show that \begin{equation} \sum_{\substack{k \geq 1 \\ k \text{ odd}}} k e^{-k^2 a} \sin(kx) \geq 0 \qquad \text{for all } x \in [0,\pi], \, a > 0. \end{equation} How should I start? I ...
0
votes
0answers
28 views

Prove that $g_n \to g$ uniformly on $\mathbb{R}$ but $f_{n}g_n$ foes not converge uniformly to$fg$ on $\mathbb{R}$

Let $f_n(x) = x$ and $f(x) = x$ for $x \in \mathbb{R}$ and let $g_n(x) = \frac{1}{n}$ and $g(x) = 0$ for $x \in \mathbb{R}$. Prove that $f_n \to f$ uniformly on $\mathbb{R}$ and $g_n \to g$ uniformly ...
0
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0answers
26 views

Continuity of convergence vector

Consider a continuous mapping $f: \mathbb{R}^n \rightarrow \mathcal{K} \subset \mathbb{R}^n$, where $\mathcal{K}$ is a compact set. Let $\alpha \in [0,1]$ and, for all $k \geq 0$, define $a_i := ...
0
votes
3answers
31 views

Calculate limit of a seriess

Calculate $$ \lim_{n \to \infty} \sqrt{n}\sin (\frac{1}{\sqrt{n}}) $$ or prove it doesn't exist.
1
vote
2answers
39 views

Simplifying a sequence formula

I wonder if this formula can be simplified (so that will be no sum symbol): $$a_n = n!\sum _{k=0} ^{n}{1 \over k!} $$ If you have any ideas utalizing generating functions or Z transform, post please
0
votes
0answers
36 views

Total Time for a ball that bounces from a height of 8 feet and rebounds to a height 5/8 [duplicate]

I am trying to solve the following problem. Let's say a ball is dropped from a height of $8$ feet and rebounds to a height $\frac{5}{8}$ of its previous height at each bounce keeps bouncing ...
1
vote
1answer
47 views

Proving $\lim \limits_{n \to \infty} b_n$ exists

If $\frac{b_n}{b_{n+1}} = 1 + \beta_n,\ n= 1,2,...,$ and the series $\sum_{n=1}^{\infty} \beta_n$ converges absolutely, then the limit $\lim \limits_{n \to \infty} b_n = b \in \Bbb R$ exists For ...
0
votes
2answers
46 views

Cauchy sequences are bounded?!

I'm having trouble understanding the proof that Cauchy sequences are bounded, here's the proof I've been given Let $s_n$ be a Cauchy sequence. We take a concrete value of $\varepsilon$, for ...