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Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: recurrence relations, convergence tests, identifying sequences, identifying terms. For questions on finite sums, use the (summation) tag instead.

2
votes
2answers
15 views

Is the sequence $( d(x_{n},y_{n}))$ convergent if $X = \mathbb{R}$ with standard topology. [duplicate]

Let $(X,d)$ be a metric space and $ ( x_{n} )$ , $ ( y_{n} )$ convergent sequences in $X$. Is the sequence $( d(x_{n},y_{n}))$ convergent if $X = \mathbb{R}$ with standard topology. I am having ...
2
votes
1answer
53 views

Is there a mathematical validity of my claims?

I have a question which is not homework. Actually, I have a hard time asking the question. But I will try to express the question as clearly and clearly as I can. In the question, since I cannot use ...
-3
votes
1answer
16 views

someone could help me know if this series is convergent or divergent.

I need help with this serie i post the image thanks for all.the_serie
4
votes
2answers
118 views

Find : $\int_0^{\pi/4}x\ln(\sin x)\mathrm dx$

I'm try to find this integral $$\int_0^{\pi/4}x\ln(\sin x)\mathrm dx$$ My try use : $\ln(\sin x)=-\ln2-\sum\limits_{n=1}^{\infty}\frac{\cos (2nx)}{n}$ But I don't know how to complete summation ......
2
votes
2answers
34 views

What does interval of convergence for power series show?

For the simplest case, $f(x)=\dfrac1{1-x}$. This can be represented by $\sum\limits_{i=1}^nx^{n-1}$ or $\sum\limits_{i=0}^nx^n$. The series is only equal to that value for $|x|< 1$, so the interval ...
0
votes
1answer
25 views

How would the Dirichlet Test for convergence prove that $\sum_{n=1}^{\infty}\frac{\cos{n}}{n}$ does in fact converge?

I've been looking for a way to determine whether $\sum_{n=1}^{\infty}\frac{\cos{n}}{n}$ converges, and the test that I've most often seen recommended seen is the Dirichlet test for convergence. ...
0
votes
1answer
18 views

Integral test with $f$ negative and increasing?

The Integral Test states Assume $f$ is continuous, positive, and decreasing on [$1, \infty$). If $\int_1 ^{\infty}f(x)\,dx$ exists and is finite, then $\sum f(n)$ converges and vice versa. ...
0
votes
0answers
17 views

Limit comparison test vs quotient test

My lecturer talks of a 'Quotient Test': Let $$s=\sum_{n=0}^∞ a_n, S=\sum_{n=0}^∞ b_n$$ Consider $$p=\lim_{n\to ∞}(\frac{a_n}{b_n})$$ If $$0\leq p<∞$$ and S converges, then s converges. $$p=∞$$ ...
3
votes
0answers
24 views

Maximum and minimum of $\frac{1}{n} \cot(n \pi \phi)$, $\phi$ Golden ratio

Studying aspects of the problem https://math.stackexchange.com/a/3186019/198592 I stumbled on this question. Designating the golden ration by $\phi=\frac{1+\sqrt{5}}{2} \simeq 1.61803$ and letting $a(...
5
votes
1answer
67 views

$\frac1n\sum _{k=1}^na_k\to0$ if and only if $\frac1n\sum _{k=1}^na^2_k\to0$

If $(a_n)$ is a sequence in $(0,1)$, show that $\frac1n\sum _{k=1}^na_k\to0$ if and only if $\frac1n\sum _{k=1}^na^2_k\to0$ My try: $\implies$: Since $a_k\in (0,1)$, we have $0\le\frac1n\sum _{k=1}^...
0
votes
1answer
34 views

Show that limit of $n(1-(\frac{2n+1}{2n+2})^p)$ as $n\to\infty$ is $\frac{p}{2}$

For proving the convergence of the following series $$\frac{1}{2}^p+\frac{1\times3}{2\times4}^p+\frac{1\times3\times5}{2\times4\times6}^p+.... \text{ for } p>2$$ I try to use Rabe test to ...
-1
votes
1answer
23 views

interval of convergence of $\sum_{n=0}^{\infty} (nx)^{n}/n! $

Im not sure how to go about finding the interval of convergence of $\sum_{n=0}^{\infty} (nx)^{n}/n! $ I think i remember learning that you can either use the ratio test or cauchy root test to solve ...
0
votes
2answers
22 views

finite geometric series ,2 different formulas?

Are both formulas the same?Do they have different use?Is there something else i am missing here? I do not ask for the proof of the formulas. i see both of them online on various resources, for ...
-1
votes
2answers
29 views

Find the number pattern [on hold]

Goal: find 1 equation to generate -4,2,10,6,3 on looping Output -4 on 1st iteration Output 2...
0
votes
1answer
13 views

Convergence of infinite series in PDE

If $$u(x,t)=\sum_{k=0}^\infty \frac{1}{(2k)!}x^{2k}\frac{d^k}{dt^k}e^{\frac{-1}{t^2}}$$ with $x\in \mathbb R$. How do is show that $u(x,0)=0$ for $x\in \mathbb R$. I know that the as $t\rightarrow 0$,...
1
vote
1answer
53 views

Value of $(\log_b2)^0(\log_b5^{4^0})+(\log_b2)^1(\log_b5^{4^1})+(\log_b2)^2(\log_b5^{4^2})+\cdots$

Find the value of $$(\log_b2)^0(\log_b5^{4^0})+(\log_b2)^1(\log_b5^{4^1})+(\log_b2)^2(\log_b5^{4^2})+ (\log_b2)^3(\log_b5^{4^3})+\cdots +\infty ,$$ where $b=2000$. My thinking: Since given Series is ...
1
vote
1answer
33 views

Determine interval of convergence for $\sum_{n=1}^{\infty} {n!(8x-1)^n}$

We have the infinite series: $\sum_{n=1}^{\infty} {n!(8x-1)^n}$ First i applied the ratio test giving $\lim_{x\to \infty} \vert\frac{(n+1)!(8x-1)^{n+1}}{n!(8x-1)^n}\vert$ After simplification: $...
-1
votes
0answers
34 views

Studying the uniform convergence of $S_n(z) = 1+z^2 - \frac{1}{(1+z^2)^{n-1}} $ [on hold]

I am trying to study the uniform convergence of $$S_n(z) = 1+z^2 - \frac{1}{(1+z^2)^{n-1}} $$ Any hint would be appreciated. Regards.
-2
votes
0answers
46 views

Relationship between $n$ and $P_{n}$

We often hear it said that either there is no relation between the natural, or counting numbers, $n$, and their counterparts the primes, $P_{n}$, or that if there is, it is so recondite as to be ...
1
vote
1answer
24 views

Hurwitz zeta function for $s=0$ $\zeta(0,1/2)$

I'm studying the Casimir Effect with perfect spherical boundary which involves the use of the Hurwitz zeta function. I've been staring for a while at this equation: \begin{align} \sum_{l=1}^{\...
0
votes
2answers
37 views

Convergence of power series beyond radius of convergence?

In rudin analysis text book, 3.44 theorem , it says about the convergence on the boundary of the circle of the power series, the theorem is roughly:- Suppose the radius of convergence of $\Sigma ...
4
votes
2answers
47 views

Prove that when writing up all even numbers in a column, then chaining $2n+1$ from them, we get all natural numbers exactly once.

Here is the idea: Write up all even numbers in the first column, then get numbers in the second column by taking the number to the left ($n$), and calculating $2n+1$. And keep repeating this. Here is ...
4
votes
2answers
55 views

$\lim_{\epsilon\to0}\frac{\cos(\epsilon-n\frac{\pi}{2})}{\epsilon^n}$

We were doing generalized integrals in class and this integral came out. I tried using integration by parts and got something repeating. We're gonna let $\epsilon \rightarrow 0$ and $x\rightarrow\...
8
votes
0answers
106 views

Triple sum $\sum\limits_{a=1}^{\infty} \sum\limits_{b=1}^{\infty} \sum\limits_{c=1}^{\infty} \frac{\cos a \cos b \cos c}{a^2 + b^2 + c^2}$

We have poor water heating system in our countryside house (currently it takes 4 hours to warm up the water), and my father has decided to improve it; he bought a water tank and placed it up in the ...
0
votes
0answers
31 views

For an increasing function $f$, if $f(y_n)-f(x_n) \to 0$ , then $f$ is continuous at $0$

Question: Let $f: \mathbb{R} \to \mathbb{R} $ be an increasing function. Suppose there are sequences $(x_n)$ and $(y_n)$ such that $x_n<0<y_n$ for all $n\geq 1$ and $f(y_n)-f(x_n) \to 0$ as $n \...
0
votes
1answer
50 views

Convergence of $\sum_{n=2}^{\infty}\left(\frac{\log n}{\log(\log n)}\right)^{-\log n/\log(\log n)}$

Initially, I need to prove, that $$\forall \lambda > 0 \ \forall\, \xi_i \sim \text{Pois}(\lambda), \xi_i\text{ are independent } \implies P\left(\limsup\limits_{n \rightarrow \infty}\frac{\...
0
votes
1answer
34 views

Let $g_n(x)=\sum_{k=1}^n (-1)^k f_k(x) \forall x\in \mathbb R. $ Then which one of the following are correct answers?

Suppose that $\{f_n\}$ is a sequence of continuous real-valued functions on $[0,1]$ satisfying the following: (A)$\forall x\in \mathbb R,\{f_n(x)\}$ is a decreasing sequence. (B)the sequence $\{f_n\}...
1
vote
2answers
40 views

How to find $A_n$ such that $\sum_{n=1}^\infty A_n\sqrt{2} n \sin (nx)=1$

I meet a trouble to find $A_n$ such that the following equality holds. $$\sum_{n=1}^\infty A_n\sqrt{2} n \sin (nx)=1, \ \ \ \ 0<x<\pi$$ I am not sure if I can really find such $A_n$ since the ...
2
votes
1answer
31 views

Proving the summation of a double factorial infinite series.

$$ \sum_{n=0}^{\infty }\frac{(-1)^{n}((2n-1)!!)^2}{(2n)! (2^{2n})} = \frac{2}{\sqrt{5}} $$ I came across this summation through some other work, came across the solution as part of a function, but I ...
-2
votes
1answer
56 views

Is there a better way to calculate the Arc-Cosine-Hyperbolic

I am using $\operatorname{arcosh} x = \operatorname{arcsinh}(\sqrt { x^2-1 } ) $ with bad results. The standard power series $$ \operatorname{arcosh} x = \ln(2x) - \left( \left( \frac {1} {2} \...
8
votes
0answers
74 views

About $\sum\limits_{r=1}^{k}\exp\left[2i\pi\sum\limits_{k=1}^{n}\frac{r^{k}}k\right]$

Context: I recently saw user @David's profile picture and description: "My icon is the graph of the exponential sum $$\sum_{n=1}^{10620}e^{2\pi if(n)}$$ for $$f(n)=\frac{n}{20}+\frac{n^2}{9}+\frac{...
0
votes
3answers
31 views

Prove that no function $f : Z → {1, …, 100} $ is one-to-one.

This seems obvious to me, but I'm not sure how I would prove it. Is simply proving $|Z| > |{1...100}| $ sufficient? If so, how would I go about proving that? I know Cantor's theorem that says some ...
1
vote
1answer
32 views

Infinite solutions to a function.

I have rewritten this question. Take a series, $$ S_k = \sum_{n=1}^k u_n $$ where $S_k \in \mathbb{C}$ forms a divergent series as $k \to \infty$. Now take a function $f_n(x)$ such that there are ...
1
vote
1answer
25 views

Finding the first three nonzero terms in the Maclaurin series: $y=\frac{x}{\sin(x)}$

As the title says I would like to find the first three nonzero terms in the Maclaurin series $$y=\frac{x}{\sin(x)}$$ I have the first few terms for the expansion for $\sin(x)=x-\frac{x^3}{6}+\frac{x^...
0
votes
1answer
28 views

Infinite series question involving integrals

Can someone help me solve this tricky infinite series problem? I tried to find the indefinite integral of the nth term but my solution didn't make sense at all. I suspect there must be an ...
3
votes
2answers
28 views

Infinite divergent series - does the convergence of quotient imply convergence of difference?

I have two series with all terms positive, $$\sum_{i=1}^{n}a_i \equiv A_n >0,\;\; \sum_{i=1}^{n}b_i\equiv B_n>0.$$ Each series diverge as $n\to \infty$. We also have $A_n \leq B_n, \forall n$. ...
1
vote
1answer
29 views

Absolute value theorem for sequences

In the proof for this theorem, my textbook states: $$-|a_n|\le a_n\le |a_n|$$ for all n. And uses this as the basis for the rest of the proof. However, I cannot seem to get the intuition. For ...
3
votes
2answers
46 views

Sum and Product inversion

Under what conditions on $a_{i,j}$ when $i\in\{1,2...,n\}$ and $j\in\{1,2...,m\}$ this relation holds: $$\sum_{i=1}^{n}{\prod_{j=1}^{m}{a_{i,j}}}= \prod_{j=1}^{m}{\sum_{i=1}^{n}{a_{i,j}}}$$ Addendum ...
1
vote
0answers
47 views

Does this explicit formula for the prime-counting function $\pi(x)$ converge?

This question is related to an answer I posted earlier at the following link. Explicit Formula for $\pi(x)$ A potential explicit formula for the fundamental prime-counting function $\pi(x)$ is ...
0
votes
2answers
35 views

Find the sum of the infinite series in Calculus [on hold]

How do I find the sum of the following infinite series? $$11 + 2 + \frac{4}{11} + \frac{8}{121} + \cdots$$
0
votes
3answers
44 views

Does $\sum_{i=2}^\infty \frac{1}{(\ln(n))^2}$ converge or diverge?

I've tried, the limit comparison test with several values and have tried finding some values for the direct comparison test but nothing really concrete has come out of it. $$\sum_{i=2}^\infty \frac{...
0
votes
2answers
71 views

Solve functional equation $f(z)=c+zf(z^2)$ with series expansion?

Let the functional equation $(1)$ be given as $$ f(z)=c+zf(z^2) \tag{1}$$ where $c \in\mathbb R$ and $c \neq 0$. How can this functional equation be solved with series expansion (power, Taylor or ...
5
votes
3answers
252 views

Find general formula for the terms

Find a general formula for the terms of the sequence $${a_n}=\left\{ \frac{11}{7},\frac{107}{49},\frac{659}{343},\frac{4883}{2401},\frac{33371}{16807},\frac{234569}{117649},\dots \right\}$$ I ...
0
votes
1answer
27 views

A specific and interesting recurrence relation

Let f and g be increasing functions such that the sets {f(1),f(2),...} and {g(1),g(2),...} partition the positive integers. Suppose that f and g are related by the condition g(n)=f(f(n))+1 for all $n&...
1
vote
1answer
55 views

Prove that this sequence of continued fractions $\frac{2}{1}, \frac{6}{5+\frac{4}{3}}, \frac{12}{11+\frac{10}{9+\frac{8}{7}}},\dots$ tends to $1$.

The Problem: I'll write up a couple more terms: $$\frac{2}{1}, \frac{6}{5+\frac{4}{3}}, \frac{12}{11+\frac{10}{9+\frac{8}{7}}}, \frac{20}{19+\frac{18}{17+\frac{16}{15+\frac{14}{13}}}}, \frac{30}{29+\...
0
votes
1answer
57 views

How do I eliminate the other wrong answers?

Let $\{u_n\}$ be a sequence of real umbers satisfying the following conditions: (1)$(-1)^nu_n\ge 0$ (2)$|u_{n+1}|<\frac{|u_n|}{2}$, for all $n\ge 13$. Which of the following statements ...
1
vote
1answer
79 views

Let $n_1,\ldots,n_k$ be positive integers summing to $N$. What's an upper bound for $\sum_{i=1}^k1/\sqrt{n_i}$?

Disclaimer. Sorry, I haven't looked into this one in any detail (as I should have). I was just thinking there out-of-be an elementary principle out here (pigeon-hole, Cauchy-Schwarz, Jensen, etc.). ...
0
votes
0answers
30 views

Rewrite a cubic summation [on hold]

how do you write $$\left(\sum_{i=1}^{k}{ i^3}\right) + (k+1)^3$$ as a single summation?
1
vote
1answer
48 views

Give a closed formula for the recursive series of $S_1 = \frac{a_1}{b_1}, S_{n+1} = \frac{a_{n+1}}{b_{n+1}+S_n}$

The Problem: The following real numbers are given: $$a_1,a_2,a_3,...\in\Bbb{R}\backslash\{0\} \\ b_1,b_2,b_3,...\in\Bbb{R}\backslash\{0\}$$ We define a recursive series of: $$S_1 = \frac{a_1}{b_1} ...
0
votes
1answer
48 views

Can we find or construct a decreasing subsequence in $(x_{n})_{n≥1}$ converging to $0$

Let $(x_{n})_{n≥1}$ be a real sequence converging to $0$. My question: Can we find or construct a decreasing or increasing subsequence in $(x_{n})_{n≥1}$ converging to $0$.