Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: identifying sequences, identifying terms, recurrence relations, $\epsilon-N$ proofs of convergence, convergence tests, finding closed forms for sums. For questions on finite sums, use the (summation) tag instead.

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36 views

How to solve this series?$\sum_{n=1}^{\infty}({\frac{1}{n^4}}\sum_{i=1}^{n}{\frac{1}{i}})$ [closed]

I only know the answer is $3\zeta(5)-\zeta(2)\zeta(3)$, but there is no detailed progress. $\sum_{n=1}^{\infty}({\frac{1}{n^4}}\sum_{i=1}^{n}{\frac{1}{i}})$
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10 views

Lower Semi Continuous Function Multiplied by a Bounded Positive Sequence.

Let $A:\mathcal{P}_2(\mathbb{R})\to \mathbb{R}$, be a lower semi-continuous functional on the space of Borel probability measures over the real line. Lower semi-continuous here means with respect to ...
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0answers
24 views

A third term of a geometric progression, which has both positive and negative terms, is 1. The sum of its first three terms is 3. What's fourth term?

for clarity, I'll put the question down again due to lack of space The third term of a geometric progression, which has both positive and negative terms, is 1. The sum of its first three terms is 3. ...
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1answer
63 views

Finding the values of $a$, $b$ and $c$ for which this series converges, but not absolutely

I want to determine the values of $a$, $b$, and $c$ for which series $$\sum_{n=3}^{\infty}\frac{a^n}{n^b(\ln(n))^c}$$ converges, but not absolutely. I tried using the Ratio Test $$\frac{a_{n+1}}{a_n}= ...
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0answers
34 views

Sequence of positive integers, $a_{n+1}$ is equal to the smallest positive integer which hasn't appeared and is noncoprime with $a_n$.

Consider a sequence of positive integers $a_1, a_2, \ldots$ such that $a_1 = 1, a_2 = 2, a_{n+1}$ is equal to the smallest positive integer which has not yet occurred in the sequence and is not ...
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0answers
14 views

Showing that the Error Reduces as the Number of Terms in a Taylor Approximation Increases

I have always had this basic question about Taylor Approximations, but I never asked it before because I always thought it was too obvious. At this point, the way we are (briefly) introduced to Taylor ...
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0answers
21 views

The difference between sum of reciprocals of prime and reciprocal of subset of composite numbers

Let $c_j$ represent all composite numbers that are divisible by primes but not by their subsequent powers, for example, 6,15 but not 8 (divisible by $2^3$). Then what is the value of $$\sum_{i=1}^n \...
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0answers
18 views

How to choose the $c$ term when evaluating the error of a Taylor series

I was looking at this link (https://personal.math.ubc.ca/~CLP/CLP1/clp_1_dc/ssec_taylor_error.html) which shows how to calculate the error of a Taylor series based on how many terms you wish to ...
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1answer
55 views

Method for convergence with unknown constant

I was expanding on specific telescopic series $$\sum_{n=1}^\infty \frac{1}{n(n+1)}= 1$$ $$\sum_{n=1}^\infty \frac{1}{n(n+2)}= \frac{3}{4}$$ $$\sum_{n=1}^\infty \frac{1}{n(n+3)}= \frac{11}{18}$$ It ...
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1answer
60 views

An order that lacks "infinite" transitivity?

The ordering on the real line "$<$" possesses the property that if we have a sequence $(a_n)_{n=1}^{\infty}$ such that $a_n < a_{n+1}$ and if we have a finite limit $a$, then $a_1 < ...
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3answers
45 views

The sequence of distances of a sequence with no limit from a compact set converges to zero

Let $(X,d)$ is a metric space and $U \subset X$ is an open set such that $X- U$ is compact, where $X-U:=\{x \in X: x \notin U\}$. Let $A \subset U$ is a countable infinite subset of $U$ such that $A$ ...
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0answers
29 views

How to show for large enough n, sqrt( (n+4)/n^4 +4 ) is larger than sqrt (x/n^4)? [closed]

I know from graphing, that sqrt( (n+4)/n^4 +4 ) is larger than sqrt (x/n^4), but how do I prove it?
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2answers
42 views

From homogeneous recursive relation to Matrices and Linear Algebra

After the lesson in recursive relations in the university, I realised that we can transform a homogeneous recursive relation to a linear algebra problem. Let $f_{n}-2f_{n-1}+f_{n-2}=0$, with $f_{0}=7, ...
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1answer
28 views

Sequence $(x_n)$ of irrationals in $[0,1]$ such that sets $\{y + x_n: y \in \mathbb{Q} \cap [0,1]\}$ are disjoint.

Let $E = \mathbb{Q} \cap [0,1]$. Show there exists a sequence $(x_n)_{n \geq 1}$ in $[0,1]$ such that the sets $E + x_n = \{y + x_n: y \in E\}$ are disjoint. The sequence must consist of irrationals ...
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1answer
26 views

In an A.P , first term = 2 & sum of first 5 terms is 1/4th of sum of next five terms. Write the equation & find d.

Q: In an A.P , first term = 2 & sum of first 5 terms is 1/4th of sum of next five terms. Write the equation & find d. My solution: $S_5$ =$\frac{1}{4}$*{$S_{10}$- $S_5$}. Q says that it is ...
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1answer
7 views

Q regarding finding total sum at end of year with an increment till n-1th year.

Q: The income of a person is 3,00,000, in the first year and he receives an increase of 10,000 to his income per year for the next 19 years. Find the total mount, he received in 20 years. Currency = ...
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0answers
44 views

By using the digits 2, 3, 4 all positive integers are written in ascending form like 2, 3, 4, 22, 23, 24, .... Find the 124th number? [closed]

It is a mcq and the options are: A) 22342 B) 22222 C) 22223 D) 22232 I am practising for an olympiad.I am not understanding this...Can someone pls explain me this and give the solution if u can so
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2answers
69 views

Determine whether the series $\sum_{n=1}^\infty\left({\arctan(n)-\frac{\pi}{2}}\right) $ converge [duplicate]

I am trying to determine whether the series $$\sum_{n=1}^\infty \left({\arctan(n)-\frac{\pi}{2}}\right) $$ converges from my calculus homework. Im not sure which candidate to choose for comprasion ...
4
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1answer
68 views

Find all function $f$ such that, for any sequence $(x_n)$ Cesàro-convergent, the sequence $(f (x_n))$ is also Cesàro-convergent

Let's say that a function $f$ is Cesàro-continuous at $x_0$ iff for any sequence $(u_n)_\Bbb{N}$ whose Cesàro mean converges to $x_0$, the Cesàro mean of the sequence $(f(u_n))_\Bbb{N}$ converges to $...
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3answers
157 views

Integral $\int_0^{\pi/3}\frac{x}{\cos(x)}\,dx$

I am trying to compute the integral $$\int_0^{\pi/3}\frac{x}{\cos(x)}\,dx \tag{1}$$ Context: Originally I was trying to prove the following result: $$\sum_{n=0}^\infty\frac{1}{(2n+1)^2\binom{2n}{n}}=\...
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1answer
47 views

Prove $\sum_{k\geq1} \frac{\sin(ka)}{ka}\cos(kx)$ is convergent

Assume $a\in ]0,\pi[$. How do I prove $\sum_{k\geq1} \dfrac{\sin(ka)}{ka}\cos(kx)$ is convergent for all $x\in\mathbb{R}$? I tried writing $\sin(ka)=\dfrac{e^{ika}-e^{-ika}}{2i}$ and $\cos(kx)=\dfrac{...
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0answers
36 views

Sigma i=0 to infinity

sigma 1)$$\sum_{i=0}^{\infty}\frac{i}{4^i}$$ $Sum$ $0 \to \frac{1}{4} \to \frac{2}{4^2} \to \frac{3}{4^3} \to \frac{4}{4^4} \to \frac{5}{4^5}\to...$ 2)$$\sum_{i=0}^{\infty}\frac{i^2}{4^i}$$ $Sum$ $0 ...
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4answers
110 views

$a_{n+1} = a_n/2 + 1/a_n$ is Cauchy but has no limit in $\mathbb{Q}$

I want to show that the sequence recursively defined by $a_{n+1} = \frac{a_n}{2} + \frac{1}{a_n}, \:\: a_1=1$ is a Cauchy sequence that does not converge in $\mathbb{Q}$. My idea was to show that ...
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0answers
35 views

Asking for clarification on showing that the set of $\mu$ measurable sets is a $\sigma$-algebra

Let $F$ be a collection of $\mu$-measurable subset of a set $A$, and suppose that it is already known that for any $S \subset X$ $\mu[S] \geq \sum_{i=1}^n\mu[S\cap B_i] + \mu[S\setminus B], k = 1, 2, ...
3
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1answer
59 views

Convergence of $\sum_{n=1}^∞ P_n(x) - f(x) $, $P_n(x)$ Is the Maclaurin polynomial of $f$

We have $f(x)$ such that $f^{(n)}(x)$ exist for all $n > 0$, And let $P_n(x)$ be the Maclaurin polynomial of $f$ (of degree $n$). Can I say for sure the following series converges for every $x$ In ...
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0answers
20 views

Closed form solution of the recurrence $a_{n+1} = (1-\lambda_n) a_n + \lambda_n b_n$ for $a_n$

Let $a_n, b_n$ be sequences of non-negative numbers, $n \geq 0$. Consider the recurrence: $$a_{n+1} = (1-\lambda_n) a_n + \lambda_n b_n$$ where $\lambda_n$ is some non-negative sequence. Is there a ...
3
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2answers
77 views

$\lim_{n\to \infty} [n+n^2\log(\frac{n}{n+1})]=\frac{1}{2}$

$\lim_{n\to \infty} [n+n^2\log(\frac{n}{n+1})]=\frac{1}{2}$ How ? I can see $\lim_{n\to \infty}\log{(\frac{n}{n+1})^n}=-1$ and rest is suggesting that the limit must not be finite. Any suggestions ...
3
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1answer
91 views

How do I prove a sequence diverges to infinity?

I have been scratching my head for a couple of days on how to determine convergence/divergence of sequences. I made it to understand how to prove that a sequence converges, but still have numerous ...
5
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1answer
151 views

Given $a_n = {(1+\frac{1}{n})}^{1/n}-1$, does $\sum_{n=1}^∞ a_n$ converge?

I am a calculus student, And I'm trying to find out If the following series converges. Given $a_n = {(1+\frac{1}{n})}^{1/n}-1$ Does the following series converges? $\sum_{n=1}^∞ a_n$ My thoughts: It ...
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0answers
40 views

Can we re-write Newton's Binomial formula as a polynomial in $\ r\ $ without any problems?

Newton's Generalised Binomial theorem states that if $\ x\ $ and $\ y\ $ are real numbers with $\ \vert x \vert > \vert y \vert\ (\text{note that } \left\vert \frac{y}{x} \right\vert < 1),\ $ ...
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0answers
41 views

Given $A_n = \sum_{i=1}^n \frac{1}{\log(i) + 1} $, Determine whether $\sum_{n=1}^∞ \frac{A_n}{n^2} $ converges. [closed]

We have $A_n = \sum_{i=1}^n \frac{1}{\log(i) + 1} $ I want to determine whether the following series converges: $$\sum_{n=1}^∞ \frac{A_n}{n^2} $$ comparison test, And the other standart theorems didn'...
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4answers
67 views

Prove that the series $\sum^{\infty}_{n=1} n^{\alpha}(\sqrt[n]{3} - 1)$ is convergent if and only if $a < 0$.

I am trying to prove that $\sum^{\infty}_{n=1} n^{\alpha}(\sqrt[n]{3} - 1)$ is convergent if and only if $a < 0$ for some time now, but so far I did not come up with anything smart. Would you ...
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0answers
8 views

Convergent incremental recurrent sequence that does not involve \(n\)

We need a way to update a value by increasing it in a way that it converges. However, it should be done without keeping a counter of the number of operations, so $n$ should not be involved in the ...
1
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2answers
79 views

How is the summation being expanded?

I am trying to understand summations by solving some example problems, but I could not understand how is the second to last line being expanded? I would really appreciate if you could explain me how ...
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1answer
41 views

Show that no sequence in $E=\{\chi_A : A \subset \Bbb R \text{ discrete}\}$ will converge to $\chi_{\Bbb R}$.

Let $X=\{0,1\}^{\Bbb R}$ and each $\{0,1\}$ discrete. We can express $X$ as characteristic functions as follows $X=\{\chi_A \mid A \subset \Bbb R\}$. Show that no sequence in $E=\{\chi_A : A \subset \...
1
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1answer
59 views

Computing $A=\sum_{k=0}^{\infty}\frac{\alpha(\alpha+k\beta)^{k-1}e^{-(\alpha+k\beta)}}{k!}f^{k}$

I'm working on a problem in order statistics. I am hoping to obtain a closed form solution for the following infinite sum: $$A=\sum_{k=0}^{\infty}\frac{\alpha(\alpha+k\beta)^{k-1}e^{-(\alpha+k\beta)}}{...
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0answers
48 views

For which $a \in \mathbb{R}$ $\sum^{\infty}_{n=1} \dfrac{(-1)^n}{n^a}$ converges and for which $a$ it converges absolutely

as mentioned in the title I want to find the values of $a$ for which series $\sum^{\infty}_{n=1} \dfrac{(-1)^n}{n^a}$ is convergent and for which values of $a$ it is absolutely convergent. Is my ...
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2answers
69 views

Convergence of $\sum_{n=1}^{+\infty}\frac{1}{n\log(n!)}$ [closed]

Im trying to determine whether or not the following series converges: $$\sum_{n=1}^{+\infty}\frac{1}{n\log(n!)}$$ Any hints/tips on how to approach? Thanks.
4
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1answer
173 views

A closed form of $\int_0^1\frac{\arctan( x^3)}{1+x}\text dx$?

I've tried series summation and contour integral, but neither works. Perhaps there's something related to number theory here. The result should be $$\frac{3\pi}{8}\log2-\frac\pi6\log(2+\sqrt{3})$$but ...
0
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1answer
12 views

Convergence of the sequence of maxima of a converging sequence of functions

We are given a sequence of functions $f_n : \mathbb{R} \to \mathbb{R}$ that converge pointwise to a limiting function $f$. Each $f_n$ has a unique maximum, $x_n$, and $f$ has a unique maximum $x_0$. ...
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0answers
48 views

Prove $\sum^{\infty}_{n=1} n^{13} q^{n}$ where $q \in \mathbb{R}$ converges if and only if $q \in (-1, 1)$ using ratio test

I am trying to prove the statement above and I have found this thread in which one of the comments suggested to use the ratio test. I have come out with following solution. Is it alright? $$\lim_{n\to\...
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0answers
16 views

Inversion problem for coefficients in an infinite series

Is there a way to calculate the coefficients $a_{n}^{i,j}(x)$ (which correspond to functions defined in $\mathbb{R}$) in the following equation? \begin{eqnarray} \sum_{n\in\mathbb{Z}}a^{i,j}_{n}(x)\...
1
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2answers
50 views

If $q$ an odd prime, is the inequality $2^n(q!)^n\mid 2(nq\text{)}!,\text{ where } n=1,2,3,4,... $ true?

Because of my research work in mathematics I have been led to believe that the following is true: Let $q$ an odd prime. Then $$2^n(q!)^n\mid 2(nq\text{)}!,\text{ where }n=1,2,3,4,... $$ This result ...
0
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0answers
40 views

Is it solvable? Sequence Recursive Formula Containing $\sqrt{\sum_{k=1}^{N-1}x_k^2\sigma_k^2}$

$$x_{k-1}=[\frac{\sigma_{k+1}}{\sigma_{k}}(\frac{x_{k}-x_{k+1}}{V_{k+1}})^\gamma+\frac{\lambda}{(\gamma+1)\eta}*\frac{x_k\sigma_k}{\sqrt{\sum_{k=1}^{N-1}x_k^2\sigma_k^2}}]^\frac{1}{\gamma}*V_k+x_k$$ I ...
2
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1answer
68 views

Showing that a function in a compact metric space $X$ has a unique fixed point. [duplicate]

I am trying to prove the following result: Let $(X,d)$ be a compact metric space, and suppose that $f:X\to X$ satisfies: $$ d(f(x),f(y))<d(x,y)\,,\forall x,y\in X\,... (1)$$ Then, there is a unique ...
-2
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0answers
20 views

Can the polynomial 1+x+x^2+x^3 ... x^n whose sum is (x^n+1)/(x-1) have multiple roots? [closed]

Familiarity with finite geometric series. Good mathematical maturity.
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0answers
44 views

How do I prove the value of the following sum? [duplicate]

I need to prove that $$\sum_{k=1}^\infty \frac{(-1)^k\cos(kx)}{k^2} = \frac{x^2}{4} - \frac{\pi^2}{12}$$ assuming $-\pi\leq x\leq\pi$. I've tried writing $\cos(kx)=\dfrac{e^{ikx}+e^{-ikx}}{2}$ but I ...
0
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1answer
67 views

Convergence $\ln\left(1+\frac{1}{n^2}\right)$ [duplicate]

I´m asking for some help, since I am stuck with a proof for convergence like follows: $$\sum\limits_{n=1}^\infty\ln\left(1+\frac1{n^2}\right)$$ I tried to separate it: $$\sum\limits_{n=1}^\infty\ln\...
2
votes
2answers
42 views

Bounded and Divergent sequence with $x_{n+m}\leq (x_n+x_m)/2$

My question is: Does there exist $x_n$ ($n\geq 0$) such that $x_n$ is a bounded and divergent sequence with $$x_{n+m}\leq (x_n+x_m)/2$$ for all $n,m\geq 0$? I'm guessing that such an example does ...
-3
votes
1answer
37 views

How-to find the common difference [closed]

A particular A.P. has a positive common difference and is such that for any three adjacent terms, 3 times the sum of their squares exceeds the square of their sum by 37.5. Find the common difference. (...

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