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

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

$\sum_{n=0}^\infty r^n \sin(n\theta)$

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 dont know any tools which can solve this question.
3
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2answers
62 views

Convergence of a crazy power series

"Let $\alpha$ be a given real number, $\alpha>0$ and $\alpha \notin \Bbb{N}$. Proof that the series $$\sum_{n=1}^{+\infty}\frac{\alpha(\alpha-1)(\alpha-2)\dots(\alpha-(n-1))}{n!}x^n$$ converges for ...
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0answers
10 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 $k \geq N$ . ...
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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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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 ...
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3answers
30 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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1answer
32 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 ...
5
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3answers
106 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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2answers
43 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 $$\left(\frac {\log(n)}{\log(n+1)}\right)^{n^{2}\log(n)}$$
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2answers
23 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
39 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
21 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
29 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. $\implies$ ...
3
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2answers
346 views

Question on Riemann's rearrangement theorem

Let there be given some conditionally convergent infinite series S. Then let R be some real number, and Qk a rearrangement of S such that the sum is equal to R. Is Qk unique? In other words, is ...
4
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1answer
28 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
16 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 ...
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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 ...
2
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1answer
48 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
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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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
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 ...
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1answer
50 views
+50

Continuity on $[a,b]$ implies uniform continuity on $[a,b]$

I don't understand the step underlined in green. I understand that for any $n$ , $|f(x_n)-f(y_n)|\geq \varepsilon$ where $x_n, y_n$ satisfy the conditions given regarding a function being not ...
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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.
3
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2answers
66 views

Evaluate $\mathop {\lim }\limits_{n \to \infty } {\left( {\frac{{{a^n} + {a^{2n}}}}{{1 + a}}} \right)^{1/n}}$ where $0<a<1$

Here is my working, feel free to add comments if you see anything wrong with it, thanks! Since $0<a<1$, we have $a^{n}<a^{n}+a^{2n}<2a^{n}$, hence $a{\left( {\frac{1}{{1 + a}}} ...
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3answers
405 views

$\sum_{p \in \mathcal P} \frac1{p\ln p}$ converges or diverges?

We will denote the set of prime numbers with $\mathcal P$. We know that the sum $\sum_{n=1}^{\infty}\frac1n$ and $\sum_{n=2}^{\infty}\frac1{n\ln n}$ diverges. It is also known that $\sum_{p \in ...
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7answers
636 views

Convergent or Divergent? $\sum_{n=1}^\infty\bigl(2^{\frac1{n}}-1\bigr)$

Is $\sum_{n=1}^\infty(2^{\frac1{n}}-1)$ convergent or divergent? $$\lim_{n\to\infty}(2^{\frac1{n}}-1) = 0$$ I can't think of anything to compare it against. The integral looks too hard: ...
3
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2answers
313 views

Show if $\sum\limits_{k=1}^\infty {a_k}^2$,$\sum\limits_{k=1}^\infty {b_k}^2$ converge, their product converges too

Show if $\displaystyle\sum\limits_{k=1}^\infty {a_k}^2$ and $\displaystyle\sum\limits_{k=1}^\infty {b_k}^2$ both converge, $\displaystyle\sum\limits_{k=1}^\infty {a_kb_k}$ also converge then Show if ...
2
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1answer
163 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 ...
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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.)
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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 ...
2
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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 ...
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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16answers
3k views

Why does the series $\sum_{n=1}^\infty\frac1n$ not converge?

Can someone give a simple explanation as to why the harmonic series $$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots $$ doesn't converge, on the other hand it grows very ...
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1answer
66 views

General solution for $x$ of $C = 100/(1+aX) + 100/(1+bX)+ \cdots + 100/(1+zX)$

Please can someone help me find a general solution for $X$: $$ C = \frac{100}{(1+aX)} + \frac{100}{(1+bX)}+ \cdots + \frac{100}{(1+zX)} $$ UPDATE Its not ideal but if we make $C = 350$ would this ...
2
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0answers
13 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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3answers
102 views

Showing that any sequence in $[0,1]$ has a convergent subsequence.

One should show that any sequence in $[0,1]$ has a convergent subsequence. Now before even trying to prove it in general, I take one sample sequence, $x_n = |\sin(n)|$. I think that for this ...
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0answers
26 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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1answer
80 views

Find all natural sequences $a_n=a_{a_{n-1}}+a_{a_{n+1}}$

Find all natural sequences for which holds $$a_n=a_{a_{n-1}}+a_{a_{n+1}}$$ a) for all natural numbers $n\ge 2$ b) for all natural numbers $n\ge 3$ I tried to do something with the characteristic ...
1
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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
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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 $$B_n(1-x)=(-1)^nB_n(x).$$ ...
1
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1answer
40 views

Prove that $\lim_{n \rightarrow \infty} \int_0^1 f_n(x)dx \ne \int_0^1\lim_{n \rightarrow \infty}f_n(x) dx$

If $f_n(x)=nxe^{-nx^2}~\forall~n=1,2,\cdots$ and $x$ real, show that $$\lim_{n \rightarrow \infty} \int_0^1 f_n(x)dx \ne \int_0^1\lim_{n \rightarrow \infty}f_n(x) dx$$ Attempt: By the $Mn$ Test, it ...
3
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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 ...
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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 = ...
2
votes
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 = ...
21
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4answers
552 views

How to compute $\prod_{n=1}^\infty\left(1+\frac{1}{n!}\right)$?

Does $$p=\prod_{n=1}^\infty\left(1+\frac{1}{n!}\right)$$ have any closed form in terms of known mathematical constants? The computer says $$p=3.682154\dots$$ but I don't even know ...
2
votes
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 $3$ ...