For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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3
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2answers
91 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 $a_{2015}$?
0
votes
1answer
226 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 ...
2
votes
1answer
193 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 $d(\{x_n\},\{y_n\})=\sup\limits_{n}|x_n-y_n|+\sup\limits_{n}\sqrt{n}|x_n-...
3
votes
1answer
81 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 ...
1
vote
1answer
227 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
45 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 : https://cs.uwaterloo.ca/...
1
vote
1answer
48 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 $n$-...
0
votes
1answer
72 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
40 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
2answers
100 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
62 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 ...
4
votes
2answers
65 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 $P=\sum_{...
15
votes
1answer
382 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} - 9}}\...
2
votes
2answers
44 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
67 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.
0
votes
0answers
117 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 $\limsup_{n\to\...
5
votes
2answers
70 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 $$\lim_{m\to\infty}\...
1
vote
1answer
108 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 ...
0
votes
2answers
39 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 ...
7
votes
3answers
318 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 \limits_{n=1}^{\infty}\dfrac{a_{n}}{1+...
0
votes
1answer
39 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. $\implies$ $\frac{\frac{(2z)^{...
3
votes
1answer
68 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 can ...
0
votes
0answers
226 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 $n^...
2
votes
1answer
63 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
63 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.
2
votes
1answer
58 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 ...
3
votes
1answer
58 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
46 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
109 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
42 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
73 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.
3
votes
0answers
73 views

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

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
50 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 $0$?...
2
votes
0answers
31 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 $n$ (...
0
votes
1answer
112 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 ...
4
votes
0answers
136 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),\, s\right)=\underset{k\geq1}{\sum}...
0
votes
2answers
129 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
57 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 $\sum a_n x^n$;...
-1
votes
1answer
60 views

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

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 ...
1
vote
1answer
178 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 ...
2
votes
1answer
55 views

Finding a specific term in a repeating sequence.

Let $f(x) = \frac1{1-x}$, and define the function $f^r$ by $$f^r(x):=\underbrace{f(f(f(...f(f}_{r\text{ times}}(x))))).$$ I am asked to find to find $f^{653} (56)$. I know that there are only $3$ ...
2
votes
0answers
212 views

How to solve this recurrence relation and solving the power series

Take the following recurrence relation into account: $$ a_{n+2} = \frac{1}{(n+1)(n+2)} \sum_{k=0}^n (s_k - (k+1)a_{k+1})(n-k+1)a_{n-k+1} $$ I know that: $$ s_{2m+1} = \frac{(-1)^m}{(2m+1)!} $$ and ...
1
vote
2answers
74 views

Is there a type of number sequence that has a nth number actually have multiple answers?

I am just looking for what this type of number sequence this is called? Example: The logic of the sequence is, take the previous numbers in the sequence and add them together in every possible way to ...
4
votes
4answers
106 views

Is this a valid way to show that the recursive sequence $x_n = x_{n-1} + \frac{1}{x_{n-1}^2}$ is unbounded?

I'm working through some analysis textbooks on my own, so I don't want the full answer. I'm only looking for a hint on this problem. Rosenlicht's Introduction to Analysis asks me to prove that $x_n = ...
2
votes
1answer
31 views

Use induction to show $a_n$ is no greater than $4\log_2(\log_2(n))$

Given a sequence where $a_1 = 1$ and $a_n = 1+ a_{\lfloor\sqrt{n}\rfloor}, n\geqslant 2$. Show that $a_n \leqslant 4\log_2\log_2(n), \forall n \geqslant 3$. Here's my idea: Base case is $n=3, a_3 = ...
1
vote
1answer
159 views

Proof of Harmonic-Geometric Mean

Let $a_1$ and $b_1$ be any two positive numbers. Let $\alpha_{n+1} = \frac{2 \alpha_n \beta_n}{\alpha_n + \beta_n}$ and $\beta_{n+1} = \sqrt{\alpha_n \beta_n}$. Show that both sequences converge and ...
1
vote
0answers
23 views

Approximating ArcCos(x) without Radicals

Take $$f(x)=2x\arccos\left(\frac{x^2+d^2-1}{2xd}\right)$$ and try and find $$ I(x)=\int_{d-1}^{3}dx f(x) \sqrt{\left(\frac{x-1}{x}\right)}\left(3-x\right)^3 $$ You'll find the result is messy (see ...
3
votes
1answer
431 views

Determine the region of convergence of series of complex functions

I have this problem. Find the region of convergence of the following series of complex functions $$ \sum_{n=1}^\infty \frac{2^n}{z^{2n}+1} $$ The progress I have made so far is that when n goes to ...
-1
votes
2answers
126 views

Help me understand how to take derivative of the PDF of X~binom(n,p) with respect to p.

This is the solution I was given. My questions: Why is it summed from k=1 to x. Shouldn't it be from k=1 to n? (If not, why not?) What is happening to the first term from line 1 to line 2? When we ...
0
votes
1answer
145 views

Is there a systematic way of resolving sequences of dots in figures?

http://bibliotecadigital.ilce.edu.mx/sites/telesecundaria/tsm01g01v01/u02t04s01.html I wonder if there is a systematic way to get the formula of a sequence of dots in figures, to resolve it faster, ...