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

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Problems on sequence and series of functions

Let $a_n$ be a sequence of real numbers. Which of the following is true? a. If $\sum a_n$ converges,then so does $\sum a_n ^4.$ b.If $\sum |a_n|$ converges,then so does $\sum a_n ^2.$ ...
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2answers
49 views

Which test to choose for these series and why?

Which convergence test we need to choose for these two equations? $$\sum_{k=1}^{\infty}\frac{k}{10+k^{2}}\tag{1}$$ $$\sum_{k=1}^{\infty}\frac{1\cdot3\cdot5\cdots(2k+1)}{4^{k}\, k!}\tag{2}$$ For ...
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1answer
37 views

Growth of a sequence

Let $$a_n=\int_{\frac{\pi}{2}+n\pi}^{ \frac{3\pi}{2}+3n\pi}\frac{\cos{t}} {t} dt$$ How to show that $\left(a_{2n}\right)_{n\geq 0}$ is increasing (strictly) and $\left(a_{2n+1}\right)_{n\geq 0}$ is ...
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64 views

Prove that $\sum_{n=1}^{\infty}\frac{a_{n}}{1+a_{n}}$ converges iff $\sum_{n=1}^{\infty}{a_{n}} $ converges

prove that $\sum_{n=1}^{\infty}\frac{a_{n}}{1+a_{n}}$ converges iff $\sum_{n=1}^{\infty}{a_{n}} $ converges. I proved one direction: $\frac{a_{n}}{1+a_{n}}\leq a_n$ therefore if $\sum a_n$ ...
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1answer
82 views

How to derive this interesting identity for $\log(\sin(x))$ [duplicate]

I saw on SE that: $$\log(\sin x)=-\log(2)-\sum_{n=1}^{\infty}\frac{\cos(2nx)}{n} \phantom{a} (0<x<\pi)$$ This is an extremely useful identity, as it helps solve: $$\int_{0}^{\pi} ...
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2answers
18 views

goemetrical progression

A credit society gives out a coompound interest of $4.5\text{%}$ per annum . Peter deposits shs.$300,000$ at the beginning if each year.How much money will ha have at the begging of the $\text{four}$ ...
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276 views

Help with difficult telescoping series question

Evaluate $$\frac3{1!+2!+3!}+\frac4{2!+3!+4!}+\ldots+\frac{2012}{2010!+2011!+2012!}\;.$$ I see that the question is telescoping, but I don't know how to break it down into a form similar to that of ...
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0answers
35 views

Uniform and absolute convergence of $\frac{1}{n^2+z^2}$

Let $z \in \mathbb{C}.$ I am asked to prove that $\sum\limits_{n=0}^{\infty} \frac{1}{n^2+z^2}$ converges on the set $\mathbb{C} \backslash \{ni : n\in \mathbb{Z}\} $. And also to prove that the ...
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sequence of linearly independent vectors

Lets say that we have I linerly independent vectors $\{v_1,v_2,...,v_I\}$. And lets say that we have a sequence of vectors $\{x^k\}^k$, where $x^k=\Sigma_Ic_i^kv_i$. Lets say that the sequence of ...
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1answer
63 views

Expressing “formally” $f(x)=\frac {1}{\sqrt {1-2x}}$ as a power series

I have to express $f(x)=\frac {1}{\sqrt {1-2x}}$ as a power series and give its interval of convergence. Knowing the binomial series is as follows this should be fairly easy: $$(1+x)^{\alpha}=\sum ...
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1answer
62 views

Series and Sequences Train Question

There's a question here that put me off, it differs from the normal AP/GP questions asked. A train is travelling at $180 \text { km/h }$, $500\text { m }$ away from a train station, what is the ...
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3answers
46 views

Divergence of Summation $a_{n+1} - a_n$?

True or false: If $\{a_n\}$ is divergent, then the series summation of $\sum_{n=1}^\infty (a_{n+1}-a_n$) is divergent. I know that this can be split into two summations: $$\sum_{n=1}^\infty ...
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3answers
33 views

Finding a closed-form formula for a sequence that is defined recursively

$$a_0 = 0, a_1 = 1 \quad \text{ and } \quad a_n = a_{n-1} + 2a_{n-2}\quad \text{ for }n\geq 2$$ a) Find $a_2,a_3,a_4,a_5$ b) Find a closed form-formula for $a_n$ I found the value to be ...
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1answer
39 views

using power expansion to find limit

I am preparing for my final exam, and stuck on this question. Using power series expansion, evaluate $$\lim_{x\to 0} \frac{x\cos(x) -\sin(x)}{x^2-x\ln(1+x)}$$ I have no idea how to proceed. ...
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1answer
31 views

Is this true?: converging sequence question

Let X be a metric space, and let A ⊂ X. Suppose that {pn} is a sequence in A which converges to some point p ∈ X. True or false: i) p ∈ A′ (limit points of A) (ii) p ∈ closure(A) These are both true, ...
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1answer
33 views

power series for $\int_0^x e^{-t^2}dt$

Use a known power series expansion to find the power series representation of the integral function $g(x) =\int_0^x e^{-t^2}dt$ centered at $a=0$ My approach Note that $g'(x) = e^{-x^2}$. ...
2
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1answer
25 views

Sequences and Series Convergence and Divergence True/False

If $\{|a_n|\}$ is convergent, then $\{a_n\}$ is convergent. If the limit as $n$ goes to infinity of $a_n$ exists, then the sequence converges. The first statement is false when $a_n = (-1)^n$ for ...
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1answer
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recursive sub-sequences of sequence , one is increasing and one is decreasing to same limit -> the sequence converge?

Let $b_1=\:0$, $b_{n+1}\:=\:\frac{1}{1+b_n}$. I need to show that $\left(b_n\right)_{n\:=1}^{\infty }$ converge. I thought about dived $b_n$ to 2 sub_sequence : $b_{2n}$, $b_{2n+1}$. (i thought ...
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1answer
14 views

2 Sub-limits of sequence converge epsilon proof

consider $\left(a_n\right)_{n=0}^{\infty }$, $L_1,L_2\:\in \mathbb{R}$. $\lim _{k\to \infty }\left(a_{2k}\right)\:=\:L_1$, $\lim _{k\to \infty }\left(a_{2k-1}\right)\:=\:L_2$ . How to prove using ...
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Find the limit of function using Taylor series

Good evening, I'm somehow stuck on solving some easy exercises : $$\lim_{x\to\infty} x^{3/2}\bigl(\sqrt{x+1}+\sqrt{x-1}-2\,\sqrt{x}\bigr)$$
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1answer
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Problem 3.14(e) in Baby Rudin

If $\{s_n\}$ be a sequence of complex numbers, define its arithmetic mean $\sigma_n$ by $$\sigma_n \colon= \frac{s_0 + s_1 \cdots + s_n}{n+1} \, \, (n = 0, 1, 2, \ldots). $$ Put $a_n = s_n - s_{n-1}$ ...
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0answers
32 views

Sequence Brain Teaser [migrated]

Fill in each blank unshaded cell with a positive integer less than 100, such that every consecutive group of unshaded cells within a row or column is an arithmetic sequence.
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1answer
14 views

Is there a difference in definition between an increasing sequence and a monotonically increasing sequence?

Is there a difference in definition between an increasing sequence and a monotonically increasing sequence? In some places I see they call it just inc/decreasing and some call it monotonically ...
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2answers
56 views

How to change the limits of a summation when the index $k$ is replaced by $-k$?

Is what I am doing below correct, please assist. $$\sum_{k=-\infty}^{-1}\frac{e^{kt}}{1-kt} = \sum_{k=1}^{\infty}\frac{e^{-{kt}}}{1-kt}$$ Is this the rule on how to "invert" the limits, and does ...
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2answers
49 views

Prove that $a_n = 2^n$

Let the recurrence relation $$ a_0 = 1 \\ a_{n+1} = \frac{2 \sum_{k=0}^n a_ka_{n-k}}{n+1} $$ I need to find a close formula for this recurrence. I've noticed that $a_n = 2^n$. I tried to prove it ...
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0answers
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Showing if $\lim_{n\to\infty} a_n=L$ then $\lim_{n\to\infty} -2a_n=-2L$ using defintion

If $\displaystyle \lim_{n\to\infty} a_n=L$ then prove using the limit definition that: $\displaystyle \lim_{n\to\infty} -2a_n=-2L$. From the given and the definition we know that: ...
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4answers
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Question about proving $\displaystyle\lim_{n\to\infty} n=\infty$ using the limit definition for a converging sequence

Prove $\displaystyle\lim_{n\to\infty} n=\infty$ using the limit definition for a converging sequence. What I did: Suppose by contra position that $n$ tends to a finite real limit $L$, so from ...
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2answers
67 views

Prove that as $x\to\infty $, $\sum\limits_{p \leq x} \frac{1}{p \log \log p} \approx \log \log \log x$

Prove that as $x\to\infty$, $$\sum_{p \leq x} \frac{1}{p \log \log p} \approx \log \log \log x$$ Here sum is taken over primes.I tried to use the partial summation formula but could not ...
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study series based on geometric series

Let ${\displaystyle S_{n}=\sum\limits_{k=0}^{n}\dfrac{1}{3^k}}$ and ${\displaystyle S'_{n}=\sum\limits_{k=0}^{n}\dfrac{k}{3^k}}$ Show that $(S_{n})_{n}$ is convergent and calculate its ...
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0answers
20 views

Randomness of a linear congruential genarator

I am working on a school project and it requires a simple pseudo-random number generator. I thought of using a linear congruential generator for this purpose. Here's a link ...
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2answers
41 views

For what sequences $a_n$ does the sequence $(1+\alpha a_n)^n$ converge?

We know $ (1+\alpha/n)^n \rightarrow e^{\alpha} $ when $n\rightarrow +\infty$. Suppose we are given a modified version of the problem: $$ \quad (1+\alpha\cdot a_n)^n \tag{1} $$ The question ...
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1answer
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Find the supremum and infimum of $A = \left \{ \frac{(-1)^{n}n}{(-1)^{n + 1} n + 1}, n \in \mathbb{N}\right \}$

$$A = \left \{ \frac{(-1)^{n}n}{(-1)^{n + 1} n + 1}, n \in \mathbb{N}\right \}$$ So the supremum is $\frac{-1}{2}$, and the infimum would be $-1$, right? However the solutions say that the infimum = ...
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1answer
67 views

How to calculate the sum of an infinite series [duplicate]

How do you calculate the sum of an infinite series like $$ \sum_{n = 0}^\infty \frac{n}{2^\sqrt{n}}$$ //EDIT //Ignore I searched up how to find this with infinite geometric series solution which ...
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2answers
126 views

$\displaystyle\lim_{n\to\infty}a^{n}\left(\frac{n+1}{n}\right)^{n^{2}}$ converges for $a$ in what range?

$\displaystyle\lim_{n\to\infty}a^{n}\left(\frac{n+1}{n}\right)^{n^{2}}$ converges for $a$ in what range? I tried $\displaystyle\lim_{n\to\infty}\ln ...
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1answer
57 views

Behaviour of the sequence $u_n = \frac{\sqrt{n}}{4^n}\binom{2n}{n}$

Let $n\in \mathbb{N}^{*}$, let ${\displaystyle u_{n}={2n \choose n}\sqrt{n}\ 4^{-n}}$ Show that $(u_{n})_{n}$ is convergent and ${l.e^{-\frac{1}{8n}}<u_{n}<l}$ The original text i'm ...
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3answers
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convergence of infinite series $\sum_{n=1}^\infty \frac{x^n}{(1+x)(1+x^2)(1+x^3)\cdot\cdot\cdot (1+ x^n)}$

I am reviewing for my final exam, and viewed this question: Decide whether the following infinite sum is convergent for all $x >1$: $$\sum_{n=1}^\infty ...
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convergence of the series $\sum_{k=0}^{\infty}E_k(x)$

Define $E_0(x)=x$, $E_1(x)=e^{E_0(x)}=e^x$, $E_2(x)=e^{E_1(x)}=e^{e^x}$, $\cdots$, $E_{n+1}(x)=e^{E_n(x)}$. For which values of $ x $ the series $\sum_{k=0}^{\infty}E_k(x)$ converges?
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1answer
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Proving by defintion: $\lim_{n\to\infty}\frac {1-n^2}{2n^2+n+5}=-\frac 1 2$

Prove by definition that: $\displaystyle\lim_{n\to\infty}\frac {1-n^2}{2n^2+n+5}=-\frac 1 2$ Scratch work to find $N\in \mathbb R$: $|\frac {1-n^2}{2n^2+n+5}+\frac 1 2|=\frac ...
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2answers
51 views

Finding the limit of this sequence using the squeeze theorem?

The n-th term is given by $$a_{n}:=\frac{n}{\sqrt[3]{n^{6}-1}}+\frac{n}{\sqrt[3]{n^{6}-2}}+...+\frac{n}{\sqrt[3]{n^{6}-n-2}}$$ then using the squeeze theorem I find that: ...
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1answer
138 views

What is $\sum_{n=1}^{\infty} \frac{n}{2^{\sqrt{n}}}$?

What is the value of $$\sum_{n=1}^{\infty} \frac{n}{2^{\sqrt{n}}}$$ It's clearly convergent but is it possible to calculate the sum?
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Study the convergence of $\sum_{n=1}^{\infty}{\prod_{k=1}^{n}{\sin (k)}}$

Can you help me to study the convergence of the following series: $$\sum_{n=1}^{\infty}{\prod_{k=1}^{n}{\sin (k)}}$$ Thanks.
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1answer
14 views

Differentiation equations and Series Expansion

When progressing through a worksheet for series I've stumbled across these form of problems which seem to boggle my mind. It does seem like it requires Maclaurins series however the approach seems to ...
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2answers
38 views

two recursive sequences and the limit of their quotient

The sequences $\left \{ a_{n} \right \}$ and $\left \{ b_{n} \right \}$ are defined by the following recurrence relations: $a_{1}=b_{1}=1$ $a_{n+1}=a_{n}+2b_{n}$ $b_{n+1}=a_{n}+b_{n}$ What ...
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4answers
57 views

A problem with proving using defintion that $\lim_{n\to\infty}\frac {n^2-1}{n^2+1}=1$

Prove using the definition that: $$\displaystyle\lim_{n\to\infty}\frac {n^2-1}{n^2+1}=1 $$ What I did: Let $\epsilon >0$, finding $N$: $\mid\frac {n^2-1}{n^2+1}-1\mid=\mid\frac ...
4
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1answer
59 views

Recurrent problem about polynomials

Given is a sequence of polynomials $P_n$, defined as follows: $P_0(x)=0, P_{n+1}(x) = P_n(x) + \frac{x-P_n^2(x)}{2}. $, n= 0,1,2,..., and x is real. Proving that for all non-negative integers n and ...
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2answers
56 views

Two series with $a_n/b_n\to 1$ converges simultaneously?

For two series $\sum a_n$ and $\sum b_n$, if $a_n/b_n\to 1, (n\to\infty)$, can we assert that if $\sum a_n$ converges, then $\sum b_n$ converges; if $\sum a_n$ diverges, then $\sum b_n$ diverges. I ...
4
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0answers
68 views

Does $\sum_{n=0}^N{1\over\sqrt{n!}}$ have a closed form?

Just out of curiosity, does the sum $$\sum_{n=0}^N{1\over\sqrt{n!}}$$ have a closed form for $N<\infty$ or eventually $N\to\infty$ ? I cannot find it anywhere and it does not resemble any function ...
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1answer
29 views

Simple question about recursive sequence format regarding $a(n+2) = -4a(n+1) + 5a(n)$

Suppose there's a recursive sequence $a(n+2) = -4a(n+1) + 5a(n)$ How can i convert it into the form $a(n)$ because I am most comfortable solving questions in this form. I tried to find out but I'm ...
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0answers
42 views

Other Patterns in Triples

I have the following 20 triples generated by polynomial distribution: $$\begin{matrix} (2,4,5)&(2,3,4)&(2,3,5)&(1,4,5)&(2,2,4)\\ ...
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1answer
37 views

Calculus II: Radius of Convergence

I have this math problem that states: In each part, write out the first four terms of the series, and then find the radius of convergence. $$(a) ...