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

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0
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2answers
24 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. $\implies$ ...
4
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1answer
27 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
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0answers
13 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 ...
2
votes
1answer
46 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
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2answers
37 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
vote
1answer
48 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
2answers
37 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
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2answers
23 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 ...
9
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1answer
47 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
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2answers
37 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
55 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.
1
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0answers
39 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
49 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
43 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
12 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 ...
0
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0answers
23 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
1answer
13 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
31 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
28 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 ...
1
vote
0answers
33 views

prove the monotonicity of a sequence and characterize its limit

I have a sequence $\{a_n\}_{n=0}^{\infty}$, which has the following recursive expression. \begin{equation*} \begin{aligned} &a_0 = p_0\\ &a_n = ...
2
votes
1answer
22 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 $3$ ...
0
votes
0answers
16 views

How to solve this recurrence relation and solving the power series

Take the following recurrence relation into account: $$ a_{n+2} = \frac{1}{(n+1)(n+2)} \sum_{k=0}^n (s_n - (k+1)a_{k+1})(n-k+1)a_{n-k+1} $$ I know that: $$ s_{2m+1} = \frac{(-1)^m}{(2m+1)!} $$ and ...
2
votes
1answer
38 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
73 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
23 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 = ...
0
votes
1answer
16 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
15 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
3answers
52 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 ...
0
votes
0answers
23 views

Show that if $a_{i -1} + a_{i}$ is a maximum for all $2 \le i \le n$, then $\sum_{i=1}^{n} a_{i}$ is a maximum? [on hold]

In particular, I would like to show that if N_{i} is such that for given values of $N_{i+1}$ and $N_{i-1}$, $\ln\left(N_{i-1}\right) + \ln\left(N_{i}\right)$ is a maximum for each $2 \le i \le n$, ...
0
votes
1answer
70 views

BMO Round 2 question [on hold]

I need help with this BMO question: The first term $x_1$ of a sequence is $2014$. Each subsequent term of the sequence is defined in terms of the previous term. The iterative formula is ...
-1
votes
2answers
41 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
16 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, ...
0
votes
1answer
25 views

Prove that a sequence can be enumerated using Catalan numbers

This problem is taken from R.P. Stanley’s Enumerative Combinatorics. Give bijective arguments to show that sequences of $n$ $1$'s and $n$ $-1$'s in which the sum of the first $i$ terms is ...
1
vote
1answer
26 views

Confused about this definition of limit superior.

The definition I am given is as follows: Let $(x_n)$ be a real valued sequence. For each positive integer $n$, let $s_n:=\sup\{x_m:m\geq n\}$. If $(s_n)$ converges, we denote its limit by ...
0
votes
0answers
24 views

How to show that $a_n + a_{n+r} + a_{n+2r} = 0$ for all $n ≥ 0$ if and only if $3r$ is divisible by $p^k − 1$?

Let ${a_n}$ $(n ≥ 0)$ be an m-sequence in $Z_p$ of period ${p^k − 1}$ where p is a prime and $k ≥ 1$. Suppose that $r$ is a natural number with $0 < r < p^k − 1$. Show that $a_n + a_{n+r} + ...
2
votes
1answer
30 views

Monotone Convergence Theorem (for real sequences) equivalent to the Least Upper Bound Property?

Some days ago I have asked this question to which André Nicolas gave a link to this paper which contained a proof of the Least Upper Bound Axiom from Monotone Convergence Theorem via Archimedian ...
2
votes
1answer
160 views

What general function can I use to represent the next sequence: 2,2,2,2,2,3,3,2,2,2,2,2,3,3,2,2,2,2,2,3,3…?

Look at this sequence: 2,2,2,2,2,3,3,2,2,2,2,2,3,3,2,2,2,2,2,3,3... It is defined as follows: $$f(n)=\begin{cases} 3 &\text{if $n \bmod 7=6,0$}\\ 2 ...
0
votes
2answers
25 views

radius of convergence $\sum a_n z^n$

Find radius of convergence of power series $\sum a_n z^n$, where $ a_n $=number of divisors of $ n^{50}$. I thought of applying root test, but the a_n looks little trickyy.
0
votes
1answer
78 views

How does this person solve the Putnam problem?

Consider this: 2003 A1 Putnam Solution. I am only looking at A1 for Putnam 2003. The problem is here: Problem A1 2003 I would like to proceed step-by-step: I understand $ka_1 = a_1 + a_1 + ... ...
0
votes
1answer
28 views

How can I perform Partial Fractions Decomposition on a Telescoping Series involving Exponentials?

Given: $\sum_{k = 1}^\infty\dfrac{6^k}{(3^k - 2^k)(3^{k+1} - 2^{k+1})}$ The Partial Fractions Decomposition is: $\sum_{k = 1}^\infty(\dfrac{3^k}{3^k - 2^k} - \dfrac{3^{k+1}}{3^{k+1} - 2^{k+1}})$ ...
5
votes
2answers
84 views

Limit $I=\lim_{n \to \infty } \sqrt[n]{\int_0 ^1 x^{\frac{n(n+1)}{2}}(1-x)(1-x^2)\cdots(1-x^n)d x}$

Im a new participant in this mathematical forum, so this is one of that i couldn't solve it. $$I=\lim_{n \to \infty } \sqrt[n]{\int_0 ^1 x^{\frac{n(n+1)}{2}}(1-x)(1-x^2)\cdots(1-x^n)d x}$$ I've ...
1
vote
0answers
27 views

partial sum of convergent series

Let series $\sum^\infty a_n$ is convergent but not absolutly convergent. And $\sum^\infty a_n $=0. Denote $ s_k $ the partial sum $\sum_{n=1}^k a_n $, k=1,2.... then which of following ARE true. 1.$ ...
1
vote
1answer
30 views

Show that ${\{s_n}\}$ is bounded if $s_n = b_1r + b_2r^2 + … + b_nr^n$ and $0 < r < 1$.

Question: Let ${\{b_n}\}$ be a bounded sequence of nonnegative numbers and r be any number such that $0 \leq r < 1$. Define $s_n = b_1r + b_2r^2 + ... + b_nr^n$ for every index $n$. Use The ...
3
votes
1answer
35 views

Proving an identity for Bernoulli polynomials

Consider the Bernoulli polynomials $B_n(x)$ given by the expansion $$\frac{te^{xt}}{e^t-1} = \sum\limits_{n=0}^{\infty}B_n(x)\frac{t^n}{n!}.$$ I want to prove the identity $$B_n(1-x)=(-1)^nB_n(x).$$ ...
0
votes
4answers
101 views

What function can I use to represent the next sequence: 2,2,2,2,2,3,3,2,2,2,2,2,3,3,2,2,2,2,2,3,3… [on hold]

What function can I use to represent the next sequence: 2,2,2,2,2,3,3,2,2,2,2,2,3,3,2,2,2,2,2,3,3... I am looking for general solution.
-3
votes
0answers
22 views

Finding the limit of the series

Find the limit of the sequence , can somebody help ,I'm struck.
1
vote
1answer
15 views

Series Expanding a Function: Complex Answer?

I have $$ f(x)=2\arccos\left(\frac{x}{2}\right)-x\sqrt{1-\frac{x^2}{4}} $$ and my friend says that the series of this function about $x=2$ (truncated to the first term) given $x\leq2$ is ...
1
vote
2answers
52 views

Prove that $\sum_{p \in \mathcal P} \frac1{p\ln \ln p}$ is divergent

In this question I've asked to decide the convergency of the series $\sum_{p \in \mathcal P} \frac1{p\ln p}$. Now I ask you to show that the series $$\sum_{p \in \mathcal P} \frac1{p\ln \ln p}$$ ...