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

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1answer
11 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 ...
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0answers
7 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 ...
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3answers
20 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
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2answers
32 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 ...
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1answer
24 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
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2answers
47 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
47 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
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2answers
30 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 ...
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0answers
11 views

Series, and limits proof. Show $|n b - \Sigma _{k =1}^{n} b_k| \leq \Sigma _{k = 1}^{N} |b_k - b| + M(n - N)$

Can someone please verify the proofs? Anything would help. Let {$b_k$} be a real sequence and $b \in R$. a) Suppose that there are $M,N \in N$ such that $|b - b_k| \leq M$ for all $k \geq N$ . ...
3
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1answer
52 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.
5
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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 ...
1
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1answer
34 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
25 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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2answers
40 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
57 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 $$\left(\frac {\log(n)}{\log(n+1)}\right)^{n^{2}\log(n)}$$
5
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3answers
109 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 ...
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1answer
31 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. $\implies$ ...
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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
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1answer
29 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 ...
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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
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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, ...
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2answers
43 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.
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1answer
49 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
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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 ...
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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 ...
9
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1answer
52 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 $x \in \mathbb{Q}$ ?
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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
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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.
1
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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
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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 ...
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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 ...
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0answers
16 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
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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 ...
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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),\, ...
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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 ...
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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 ...
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1answer
32 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 ...
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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 ...
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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 = ...
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1answer
23 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 $3$ ...
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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 ...
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1answer
40 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
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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
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1answer
24 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 = ...
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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 ...
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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
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2answers
63 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 ...
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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
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1answer
71 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 ...
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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 ...