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

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0
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3answers
44 views

convergence of $(2^n + 3^n)^{\frac{1}{n}}$

prove that the sequence $\{(2^n + 3^n)^{\frac{1}{n}}\}$ converges and find its limit. Ive deduced that the sequence will converge to 3, just need help writing a formal proof of this fact. thanks!
1
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3answers
75 views

Sequence proof if it exists

For every integer r>=3 there exists a sequence $a_{1,\space }a_{2,}.....,a_r$ of nonzero integers with the property that $a_1^2+a_2^2+....+a_{r-1}^2=a_r^2$ I tried to prove this with proof by ...
1
vote
1answer
48 views

Proving a series diverges

Hello I am trying to prove the following series diverges $$\sum_{k=1}^{\infty} \ln\left(1+\frac{(-1)^{k+1}}{\sqrt{k+1}}\right)$$ This series alternates around 0 and goes to zero but fails the ...
-1
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1answer
22 views

Reducing algebraic summation

I am a computer programmer by trade and am studying algorithm analysis...because i am masochistic like that. Anyhow, I was looking at the solution for one of the problems in the book. However, I am ...
3
votes
2answers
64 views

How find all possible values of $a_{2015}$ for $a_2 = 5$, $a_{2014} = 2015$, and $a_n=a_{a_{n-1}}$?

Let $(a_i)_{i\in \Bbb{N}}$ be a sequence of nonnegative integers such that $a_2 = 5$, $a_{2014} = 2015$, and $a_n=a_{a_{n-1}}$ for all other positive $n$. How find all possible values of $a_{2015}$?
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1answer
17 views

How to work out following problem -> “sum of the multiples of 6 less than 100”

I'm not sure how to approach this without just brute forcing it, which would be doable for numbers lower than 100 but obviously not great and certainly not for 2000 or something. I know that the ...
1
vote
1answer
39 views

Is the unit ball in this sequence space compact?

I have a set $X=\{\text{complex sequences } \{x_n\}: \sup\limits_{n}\sqrt{n}\left|x_n\right|\leq 1\}$ equipped with a metric ...
-1
votes
1answer
31 views

Proof or confute $\limsup_{k\to\infty} a_k + b_k = a + \limsup_{k\to\infty} b_k$ [on hold]

There are two real sequences $(a_k)_{k=1}^\infty, (b_k)_{k=1}^\infty$ and i) $(a_k)_k$ is convergent with $\lim_{k\to\infty} a_k = a$ ii) $(b_k)_k$ is bounded. Prove or confute: ...
3
votes
1answer
42 views

Need help with continuing an idea concerning showing that $4\sum\limits_{n \ge 1} a_n^2 \ge \sum\limits_{n \ge 1} \frac1{n^2}(a_1+…+a_n)^2 $

I recently encountered the following problem: If $\sum a_n^2 $ converges and $\alpha_n= \frac{a_1+...+a_n}{n}$ then show that: $$4\sum_{n \ge 1} a_n^2 \ge \sum_{n \ge 1} \alpha_n^2$$ I had an ...
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2answers
38 views

Radius of convergence for $\sum_{n=1}^\infty ((n^2+3)/n^2)^{n^3}z^n$ [on hold]

What's the convergence radius of this power series? $$\sum_{n=1}^{\infty}\left(\frac{n^2+3}{n^2}\right)^{n^3}z^n$$ I think it works with the theroem of Cauchy-Hadamard. Thanks.
0
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1answer
17 views

Properties of sup and lim inf.

Let $(a_{n})_{n \geq1}$ be a sequence of numbers such that $a_n\leq M$ for all $n \geq 1$ . Prove that $$ \lim_{n\to\infty} \inf \{a_n,a_{n+1},...\} = \sup_{n \geq1} \inf\{a_n,a_{n+1},...\} $$ ...
1
vote
1answer
30 views

How to find $\int_0^{1/4}\frac{1}{x\sqrt{1-4x}}\ln\left({\frac{1+\sqrt{1-4x}}{2\sqrt{1-4x}}}\right)dx$

Let $H_n$ be the harmonic series. I want to find the value of $A=\displaystyle\sum_{n=0}^\infty \binom{2n}{n}\left(\frac{1}{4}\right)^n\frac{H_n}{n} $. From this paper : ...
1
vote
1answer
31 views

Doubt on presumably divergent series with primes

I am wondering if my reasoning is correct. I want to determine if the following series converges or not: \begin{equation} \sum_{n=1}^\infty\frac{1}{(\ln p_n)^2} \end{equation} where $p_n$ is the ...
0
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1answer
32 views

Solving the Sequence of this question on Putnam Exam

Problem: Solution: Solution for 2003 A1 Putnam $ka_1 = a_1 + a_1 ... a_1 \le n \le a_1 + (a_1 + 1) + (a_1 + 1) ... (a_1 + 1)$ $= ka_1 + k - 1$ I know these then: What should I do next? Without ...
2
votes
4answers
31 views

how to find $a_{50}$ from a recursive term

Given $a_{n+1}=a_n+2n+3,a_1=3$ How can I find $a_{50}$? I can compute $a_2,a_3,...,a_{50}$ But it's a long way. Is there any smart technique to compute? Thanks.
3
votes
3answers
55 views

What is $\limsup n^ne^{-n^{1.001}}$?

During checking whether or not $\sum_{n=1}^{\infty}{n^ne^{-n^{1.001}}}$ converges, I thought of trying the n-th root test. I got that $\sqrt[n]{n^ne^{-n^{1.001}}}=ne^{-n^{0.001}}$. How can I find ...
1
vote
1answer
35 views

Series convergence or divergence how to test

I have the following series defined. $$\displaystyle\sum_{k=1}^{n} \cos \left( {\frac{\pi}{2}} k \right) \frac{k}{k+1000} \frac{1}{\sqrt{k}}$$ where $n = 1,2...$ How to test whether this series ...
5
votes
2answers
57 views

Evaluate the sum $P=\sum_{n=1}^\infty \dfrac{a_n}{2^n}$.

Question: Let ${\{a_n}\}$ be the sequences of $0$s and $1$s, such that $a_n=1$ if $p$ is a prime number, otherwise $a_n=0$. So, ${\{a_n}\}={\{0,1,1,0,1,0,1,0,0,0,1,...}\}$. Evaluate the sum ...
6
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0answers
57 views

Gosper's unusual formula connecting $e$ and $\pi$

Wolfram MathWorld quotes (see equation $(26)$) Gosper gives the unusual equation connecting $\pi$ and $e$ $$\sum_{n = 1}^{\infty}\frac{1}{n^{2}}\cos\left(\frac{9}{n\pi + \sqrt{n^{2}\pi^{2} - ...
3
votes
2answers
33 views

Prove that $d(x,y)=\sum_{i=1}^\infty \frac{|x_i - y_i|}{2^i}$ converges

Question: Let S be the set of sequences of $0$s and $1$s. For $x = (x_1, x_2, x_3, ...)$ and $y = (y_1, y_2, y_3, ...)$. Define $d(x,y)=\sum_{i=1}^\infty \dfrac{|x_i - y_i|}{2^i}$ Proof the ...
4
votes
0answers
66 views

How to find the value of $\sum\limits_{n=0}^\infty r^n \sin(n\theta)$? [duplicate]

Question is to find the value of $$\sum_{n=0}^\infty r^n \sin(n\theta)\text{ for }r=0.5\text{ and }\theta=\pi/3$$ I don't know any tools which can solve this question.
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votes
0answers
22 views

Finding limsup and liminf for odd and even $A_n$

I am trying to understand $\limsup$ and $\liminf$. I have this homework problem: For each natural number $n$, let $A_n=[0,1]$ if $n$ is odd, and $A_n=[1,2]$ if $n$ is even. Find both ...
5
votes
2answers
31 views

Does the limit of this double sequence exist?

Consider $$a_{mn}=\frac{m^2n^2}{m^2+n^2}\left(1-\cos\left(\frac{1}{m}\right)\cos\left(\frac{1}{n}\right)\right)$$ Does $\lim_{m,n\to\infty}a_{mn}$ exist? It can be seen that ...
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vote
1answer
35 views

Understanding graphically the convergence of alternating harmonic and divergence of harmonic

I understand the rules of convergence of a series so that I know that $\sum \frac 1 n$ (the harmonic series) diverges and $\sum \frac 1 {n^2}$ squared converges. It doesn't make sense graphically to ...
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2answers
31 views

Convergence of a complex series

I have a question about this series: $$ \sum_{n=0}^\infty \left( \frac{\sqrt{3} - i}{2} \right)^n $$ How can I show whether the series converges or not? The problem is that the root test and the ...
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votes
2answers
42 views

Prove this series is convergent. [on hold]

Prove this series is convergent. $$0-\frac{1}{2}+\frac{1}{2^{2}}-\frac{1}{3}+\frac{2}{3^{2}}-\frac{1}{4}+\frac{3}{4^{2}}- ...$$
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2answers
59 views

To check convergence/divergence of $\left(\frac {\log(n)}{\log(n+1)}\right)^{n^{2}\log(n)} $ [on hold]

How do I check convergence/divergence of series whose $n$-th term is given by expression below $$\left(\frac {\log(n)}{\log(n+1)}\right)^{n^{2}\log(n)}$$
6
votes
3answers
139 views

$ \sum_{n=1}^{\infty}a_{n} $ diverges but $ \sum_{n=1}^{\infty}\frac{a_{n}}{1+a_{n}^{2}} $ sometimes converges and sometime diverges.

Let $ \lbrace a_{n}\rbrace $ be a sequence of positive terms such that $ \sum \limits_{n=1}^{\infty}a_{n} $ diverges. I am going to show that the series $$ \sum ...
0
votes
1answer
32 views

For which $z \in \mathbb c$ does this series converge?

$f(z)=\sum_1^\infty \frac{(2z)^{2k}}{2k(2k-1)}$ I didn't know how to start so I just tried the ratio test. If $|\frac{a_{n+1}}{a_n}|>0$ then the series converges. $\implies$ ...
0
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0answers
7 views

Bound from distinct integer summation

We want to find $r$ positive integers $\{a_i\}_{i=1}^r$ such that of atmost $(s+1)^r$ values obtained from $$\sum_{i=1}s_ia_i$$ where $s_i\in\{0,\dots,s-1,s\}$, we insist on some combination of ...
4
votes
1answer
30 views

Prove $f(x) = \sum_{n=1}^\infty \frac{\sin nx + \cos nx}{n^3}$ is well-defined and $C^1$.

Prove $f(x) = \sum_{n=1}^\infty \frac{\sin nx + \cos nx}{n^3}$ is well-defined and $C^1$. First of all I need to prove that $f(x)$ is well-defined. I'm not so sure what does it mean. Basically I ...
0
votes
0answers
17 views

Identifying the Dominant terms

I've got the following sequence and I'm trying to work out what the dominant term is. $$a_n=\frac{3n^2+5^n}{2^n-5}$$ now I know that the dominant term is either $5^n$ or $2^n$ as $c^n$ dominates ...
3
votes
1answer
50 views

Is $[a,\infty )$ closed

In standard topology, of course $[a,\infty )$ is closed since its complement is open. But I don't know how to prove closeness of $[a,\infty )$ in Real Analysis using just the definition of closeness, ...
0
votes
2answers
45 views

Limit of sequence $n!\left(\frac{e}{n}\right)^n$

Find the limit of $$ \lim_{n\to +\infty} n!\left(\frac{e}{n}\right)^n. $$ I have shown that $u_{n+1}>u_n$, but I am not sure where to go from here.
1
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1answer
50 views

Summing two different series

I was wondering how to sum the first n terms of the following series: $1, 1/2, 1/4, 1/4, 1/8, 1/8, 1/8, 1/8,\ldots$ $1, 1/2, 1/2, 1/4, 1/4, 1/4, 1/4, 1/8,\ldots$ I am trying to find a tight bound ...
4
votes
2answers
50 views

Proving convergance of a series

I need to determine whether the series $\sum^{\infty}_{k=1}\frac{1}{(-1)^kk +2}$ converges or disverges. Surely, it's not absolutely convergent. I tried using Dirichlet's test by multiplying numerator ...
1
vote
2answers
24 views

Finding a good comparison/bound for determining the convergence of a series

The series is defined as follows: $b_0=1,b_1=-7,b_k=2b_{k-1}+b_{k-2}$. I need to find a good comparison sequence to determine whether $\sum_{k\geq0}1/b_k$ converges. I considered using $1/k^2$, which ...
11
votes
1answer
57 views

Existence of a sequence of positive continuous functions on $\mathbb R$ which is unbounded precisely on $\mathbb Q$

Is it possible to find a sequence of positive continuous functions $g_n$ on the real numbers such that $( g_n(x) )$ is unbounded if and only if $x \in \mathbb{Q}$ ?
0
votes
2answers
38 views

If there is an $x$ such that $(2x_n)$ converges, does this imply that $(x_n)$ is also convergent?

I am toying around with the definition of convergence of sequences. Ans I asked myself the following question: Let $(x_n)$ be a real sequence such that there is an $x\in\mathbb{R}$ such that for all ...
3
votes
3answers
57 views

Check whether or not $\sum_{n=1}^{\infty}{1\over n\sqrt[n]{n}}$ converges.

Check whether or not $\sum_{n=1}^{\infty}{1\over n\sqrt[n]{n}}$ converges. I tried few things but it wouldn't work out. I would appreciate your help.
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0answers
41 views

is the sequence $[ne]$ convergence?

Is the sequence $a_n=[ne]$ convergence or partially convergence? ($e$ is the Euler's number and the bracket mean the integer part function.)
3
votes
0answers
50 views

$\lim_{n \rightarrow \infty}\frac{1}{n}\sum_1^n\frac{k^8}{(a+(k+b)^2)^4}=1$

I am just having fun with this question: Is this true that $\displaystyle \lim_{n \rightarrow \infty}\frac{1}{n}\sum_1^n\frac{k^8}{(a+(k+b)^2)^4}=1$? I thought to change this to an integral, namely ...
0
votes
0answers
45 views

Finding the limit of the series .

I am dealing with the following: $$\lim_{x\to 0}\left(\lim_{n\to\infty}\sum_{i=1}^n i^{\frac{1}{\sin^2(x)}}\right)^{\sin^2(x)}$$ What is the limit, it is of the type infinity raised to the power ...
2
votes
0answers
20 views

Relationship between asymptotic distribution and logarithmic sums of elements of subset of the natural numbers

Consider a subset $A$ of the natural numbers analogous to the primes (but rarer). Let $a_n$ denote the $n$th element of $A$, and $a(n)$ denote the number of elements of $A$ less than or equal to $n$ ...
2
votes
1answer
80 views

Show that ${\{x_n}\}$ is convergent and monotone

Question: For $c>0$, consider the quadratic equation $$ x^2-x-c=0,\qquad x>0. $$ Define the sequence $\{x_n\}$ recersively by fixing $x_1>0$ and then, if $n$ is an index for which $x_n$ has ...
1
vote
0answers
27 views

About Mellin transform and harmonic series

Let $\mathfrak{M}\left(*\right)$ the Mellin transform. We know that holds this identity$$\mathfrak{M}\left(\underset{k\geq1}{\sum}\lambda_{k}g\left(\mu_{k}x\right),\, ...
0
votes
2answers
37 views

What kinds of things are infinite series useful for?

I'm a relative beginner studying Calc II, and I haven't been very motivated while learning infinite series. Can you give some examples of how they're used in calculus or other areas of math, or in ...
0
votes
2answers
22 views

Radius of convergence powers series s.t seriex $ \sum |a_n| $diverges

Let $\{a_n:n\ge 1\}$ be sequence of real numbers such that the series $\sum a_n$ converges and the series $\sum |a_n|$ diverges. Let $R$ be the radius of convergence of the power series $\sum a_n ...
1
vote
1answer
33 views

Counterexample for “subsequence of a convergent sequence is convergent to same limit” [on hold]

Let ${\{a_n}\}=\left\{\dfrac{1}{n}\right\}$ s.t. $n\in \mathbb{N}$, and let ${\{b_n}\}=\left\{{\dfrac{1}{n}}\right\}$ s.t. $n\in {\{1,...,N}\}$. How it is possible that ${\{b_n}\}$ is a subsequence of ...
3
votes
1answer
29 views

Proof of existence of “peaks” in The Sequential Compactness Theorem

According to Bolzano–Weierstrass theorem: How do we guarantee that EVERY sequence of real numbers has either infinitely or finitely many "peaks"? (Note that a subsequent is ordered in a way of ...