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.

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2answers
32 views

Determine whether the series is converges or not.

$$\sum_{n=0}^\infty\left(\sqrt{n+\sqrt{n}}-\sqrt{n}\right)$$ The question is to determine whether the series converges or diverges. I’ve tried to use the integral test but I couldn’t figure it out.
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1answer
32 views

The integral of an infinite sum

For the integral: $\int_{1}^{\infty}\sum_{n=x}^{\infty}\frac{1}{n^{4}}dx$ How does one go about calculating this? I've only learned how to do U-substitution, and I lack the knowledge to somehow ...
0
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0answers
8 views

What does $\sum_{m=1}^n \ln (1 -\frac{\theta^2}{n}+ \frac{\theta^2}{2n m(\log m)^2} )$ converge to?

What does the following sum converge to as $n\rightarrow\infty$? $$\sum_{m=2}^n \ln (1 -\frac{\theta^2}{n}+ \frac{\theta^2}{2n m(\log m)^2} )$$ I think it converges to $-\theta^2/2$, but I am not ...
0
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1answer
15 views

Show that the series converges and find the error

I have the following question here: a) Show that $ \displaystyle \sum_{n=1}^{\infty} \left( \sin \left( \frac{1}{2n} \right) - \sin \left( \frac{1}{2n+1} \right) \right)$ converges b) Find an ...
0
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1answer
24 views

finding a formula for the generating function for a recurring sequence

I have the sequence $a_0=0$, $a_1=3$, $a_2=0$, $a_3=23$, and $a_n=6a_{n-2} + 8a_{n-3} + 3a_{n-4}$ for $n\ge 4$ and I have to find the formula for the generating function $A(t)=\sum_{n=0}^\...
0
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1answer
10 views

Characterizing the closure of metrizable spaces with monotone sequences.

Let $(X, \tau, \ge)$ be a metrizable totally ordered space and $A \subseteq X$. Then $ x \in \overline{A}$ if and only if there is a monotone converging sequence to $x$. The idea i’m following is ...
-1
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2answers
35 views

Why is this summation rule true? [closed]

I just encountered a proof in a book that used the following: $$ \sum_{i=1}^n{2^i + n} = 2^{1+n} - 1 + n $$ Why is this true? This is the part that I am refering to
0
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1answer
50 views

Is it true? If $\lim_{n\to\infty}\frac{|a_{n}|}{|c_{n}|}=0$, then the series $\sum_{n=1}^\infty a_n$ converges

Let $\{c_n\}$ be a sequence that converges to $0$ and let $\{a_n\}$ be a sequence for which the following applies: $\lim_{n\to\infty}\frac{|a_{n}|}{|c_{n}|}=0$. Then the series $\sum_{n=1}^\infty a_n$ ...
0
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2answers
18 views

Prove that if $\lim_{n\to \infty} s_n = \infty$ and if $(t_n)$ is a bounded sequence, then $\lim_{n\to \infty} s_n + t_n = \infty$

I know that since $(t_n)$ is bounded, $\exists$ an $M_1$ > 0, such that $(t_n)$ < $M_1$ and since $\lim_{n\to \infty} s_n = \infty$, $\forall$ n > 0, $\exists$ N($M_2$) such that n > N $\Rightarrow ...
1
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3answers
18 views

Rewriting an explicit formula as an recursive formula

I have the following series: $x_n = \frac{n}{n+1}$ and I'm trying to rewrite it as a recursive formula. I wrote the first 5 elements: $1/2, 2/3, 3/4, 4/5, 5/6$ however I fail to find the recursive ...
1
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1answer
21 views

First condition of Alternating Series Test, for the sequence $(-1)^n\cdot \ln(n)/n^{1/2}$?

The series $$\sum_{n=1}^\infty (-1)^n\frac{\ln n}{\sqrt{n}}$$ does not satisfy the first condition of the Alternating Series Test: $b_{n+1}$ less than or equal to $b_n$, for all $n$ (for example $n=...
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2answers
53 views

Find a generating function of $(0,0,1\cdot2^1,0,0,2\cdot2^2,0,0,3\cdot2^3,\ldots)$

Find a generating function of: $$(0,0,1\cdot2^1,0,0,2\cdot2^2,0,0,3\cdot2^3,\ldots)$$ We can write this as a generating function: $$f(x)=x^0\cdot0+ x^1\cdot0 + x^2\cdot1\cdot2^1 + \cdots$$ Which is:: $...
2
votes
1answer
85 views

Sum of finite series

The sum of series $\frac{8}{5} + \frac{16}{65} + ....+\frac{128}{2^{18}+1} $ is A) $\frac{540}{1088}$ B) $\frac{1088}{545}$ C) $\frac{1001}{500}$ D) $\frac{1013}{545}$ I am unable to figure out ...
1
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1answer
38 views

How to show that this sequence really converges

How I'm supposed to show that the sequence $x_i$ , (i is an integer number) that is given below converges to $2$. $x_0=x_1=1$ and for $i\geq1$: $$x_{i+1}=x_i+x_{i-1}-(x_i \cdot x_{i-1})/2$$ By ...
0
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3answers
31 views

Showing that this particular sequence is Cauchy using the definition

This is the following information that I've been given; $$a_{1}=1,\, a_{2}=4,\, a_{n+2}=\frac{2a_{n+1}+a_{n}}{3}$$ I need to show that $|a_{m}-a_{n}|<\epsilon \,,\forall n\geq n_{0}, m>n$ I ...
0
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0answers
24 views

Explicit expression of sequence and rationality

We have : $\phi_{1,1} := 1$, and $$\phi_{a,b} := \frac{\phi_{a-1,b-1}}{b(b-1)}$$ for $a \in [\mspace{-2 mu} [2,\infty[ \mspace{-2 mu} [$ and $b \in [\mspace{-2 mu} [2,a] \mspace{-2 mu} ]$ and we have $...
2
votes
2answers
71 views

Prove the following limit about $e$ [duplicate]

I need to show that: $$\lim_{n \to +\infty}\sum_{k=0}^{k=n-1}\left(\frac{n-k}{n}\right)^n = \frac{e}{e-1}$$I observed that taking the limit term by term gives the result, but of course this is not ...
1
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3answers
77 views

Evaluate of: ${\prod_{n=1}^{\infty}\left[1+\frac{1}{\sum_{j=1}^{n}F_j^2}\right]^{(-1)^n+1}}$

How do we evaluate this infinite product with a sum within it? $$\large{\prod_{n=1}^{\infty}\left[1+\frac{1}{\sum_{j=1}^{n}F_j^2}\right]^{(-1)^n+1}}$$ Where $F_j$ is the Fibonacci number If I open ...
-2
votes
1answer
32 views

Arithmetic and Geometric series in one

I need to know how to solve this type of problem. Its a practice problem for a math competition: Given a 4-term sequence such that the first 3 terms form an arithmetic sequence and the last 3 terms ...
0
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1answer
22 views

How to calculate the limit of a series that contains the index both as an exponent and as a base? [duplicate]

It's easy to prove whether or not, for example, $\sum_{i=0}^\infty \frac{i^2}{2^i}$ converges, but how do you calculate the limit of such a series?
2
votes
3answers
99 views

Proving $ \lim_{n \to \infty} \sum _{k=0}^{n-1} \left(1- \frac{k}{n}\right)^n= \frac{e}{e-1}$ [duplicate]

Show that $$ \lim_{n \to \infty} \sum \limits_{k=0}^{n-1} \left(1- \frac{k}{n}\right)^n= \frac{e}{e-1}$$ Since $\lim_{n \to \infty} \left(1- \frac{k}{n}\right)^n= e^{-k}$ and infinite sum of G.P. ...
0
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0answers
48 views

Finding $\lim_{n\to\infty}\frac{\log(x_n)}{n}$ for a given sequence $(x_n)$.

Given a sequence $(x_n)$ such that $x_0=1$ and $x_n=\begin{cases}2x_{n-1}+1, & n&\text{is odd}\\ 3x_{n-1}+2, & n & \text{is even} \end{cases}$ How do I show that $\lim_{n\to\infty}\...
0
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3answers
56 views

What is the value of $[(1^ {-2/3}) + (2^ {-2/3}) + (3^ {-2/3}) + … + (1000^ {-2/3})] $? Where $[x]$ stands for the greatest integer function.

What is the value of $$[(1^ {-2/3}) + (2^ {-2/3}) + (3^ {-2/3}) + … + (1000^ {-2/3})]?$$ where $[x]$ stands for the greatest integer function. P. S : I ran a code on my PC to find that the ...
1
vote
1answer
42 views

Prove $g(x) = \sum_{k=0}^\infty \frac{(-1)^k}{2k+1}\,x^{2k+1}$ converges uniformly on [-1,1]

For the problem below, can someone please tell me how to enhance my proofs? I am not confident in my proofs. Thank you! $\textbf{Problem:}$Let $\displaystyle{g(x) = \sum_{k=0}^\infty \frac{(-1)^k}{2k+...
4
votes
1answer
94 views

$1-1+1-1+1-1+\cdots$ and $1-2+3-4+5-6+7-\cdots$ and Taylor's theorem

Some background. I was exploring the series expansion for $\ln(1+\cos x)$ in an attempt to expand it (at least, initially!) up to the third non-zero term, and along the way I unexpectedly stumbled ...
-1
votes
1answer
26 views

Show that $\lim_{|z| \to \infty} |p(z)| = \infty$ for a non-constant polynomial $p$

Let $f: \mathbb{C} \to \mathbb{C}$ be a function. By definition we have $\lim_{|z|} |f(z)| = \infty$ when for every $M \in \mathbb{R}$ there exists some $R > 0$ such that $|f(z)| \geq M$ for all $z ...
2
votes
2answers
96 views

Showing that a series is convergent

So I have the following question: Show that the following series are convergent: a) $\displaystyle\sum_{n=2}^{\infty} \frac{\log_n(n!)}{n^3}$ b) $\displaystyle\sum_{n=1}^{\infty} \frac{1}{(3n-2)^{...
0
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3answers
25 views

How to determine $\text{max } |z^n + a|$ for $|z| \leq 1$, $a \in \mathbb{C}$, $n \in \mathbb{N}_{\geq 1}$?

Let $n \in \mathbb{N}_{\geq 1}$ and $a \in \mathbb{C}$, determine $\text{max } |z^n + a|$ for $|z| \leq 1$. I do not understand the question, what is it exactly that I need to determine? And how do I ...
2
votes
3answers
42 views

How to show if $\sum_{n=1}^\infty 7X_{n+1} + X_n$ converges and $X_n \to 0$ when $n \to \infty$, then $\sum_{n=1}^\infty X_n$ converges?

Note that $X_n$ is just a sequence of real numbers. I'm a bit out of ideas because I'm used to having the extra hypothesis of $X_n$ being of positive terms. Any hints or suggestions will be greatly ...
1
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2answers
77 views

Solving $T(n) = 2T(\frac{n}{4}) + \log n$

Solve following recursive relation $T(n) = 2T(\frac{n}{4}) + \log n$ without resorting to the master theorem. I've tried substitution method but it didn't work. I don't know whether there is a method ...
0
votes
2answers
31 views

Finding explicit formula of a series

I'm trying to find the explicit formula of a series: $a_1=\frac{1}{2}, a_{n+1}=\frac{(n+1)^2}{n(n+2)}a_n$ I calculated, that $,a_2=\frac{9}{6},a_3=\frac{3}{5},a_4=\frac{5}{8}$. However, I fail to ...
0
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1answer
27 views

Sequence convergent to $e$ [duplicate]

Suppose $\lim_{n\to\infty}na_n=0$. We want to show that $$\lim_{n\to\infty}\left(1+\frac{1}{n}+a_n\right)^n=e$$ I am trying to use squeeze them, using $-\frac{2}{n}<a_n<\frac{2}{n}$ but could ...
0
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0answers
29 views

Finding the limit given $X(1)$ and $X(n+1)$ [duplicate]

Given $X_1 = 0$, and $$X_{n+1} = \frac{X_n^2 + 1}{2}$$ how can I use this information to find $$\lim_{n\to\infty} X_n$$ I only really have a basic understanding of how limits work but I have no idea ...
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0answers
22 views

Prove a sequence monotone & seek limit

Let $a_n=\sqrt[n]{n},b_n=\frac{a_{n+1}}{a_n}$. Prove that from $n=5$, $(b_n)_{n\in\mathbb N}^{\infty}$ is increasing and find $\displaystyle\lim_{n\to\infty}b_n$ This is a textbook problem in chapter ...
2
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0answers
61 views

Evaluating $\int_0^1\frac{\ln x\operatorname{Li}_2(x)\ln(1-x/2)}{x}\ dx$

How to evaluate $$\int_0^1\frac{\ln x\operatorname{Li}_2(x)\ln(1-x/2)}{x}\ dx\ ?$$ I came across this integral while I was trying to calculate $\int_0^{1/2}\frac{\operatorname{Li}_2^2(x)}{x}\ dx$ ...
0
votes
0answers
12 views

Approximations for infinite sums of a sequence resembling an exponential power series

Let $\alpha,\beta,\gamma>0$ and consider a sequence $\{A_k\}_{k\in\Bbb N}$ defined by $A_0=1$ and $$A_{k}/A_{k-1}=\begin{cases}\frac{\alpha}{k\beta} & ,k\leq n\\\frac{\alpha}{n\beta+(k-n)\...
0
votes
2answers
27 views

Calculating the convergence order

I have the iterative formula: $$x_{n+1}=x_{n-1}\cdot (x_n)^2$$ How can i calculate the convergence's order when the series is not a constant but converge. I know that i have to substitute $x_n=\...
0
votes
0answers
38 views

Find whether $\sum_{n=2}^\infty\frac{1}{n\ln^2(n)}$ converges [duplicate]

I tried to compare it with $1/n$ which I know diverges: $$\frac{1}{n\ln^2(n)} \le \frac{1}{n} \Leftrightarrow \\ n\ln^2(n) \ge n \Leftrightarrow \\ \ln^2(n) \ge1\Leftrightarrow \\ \ln(n) \ge 1 \...
2
votes
1answer
41 views

If $f(r)=r(r+1)(r+2)(r+3)$, simplify $f(r+1)-f(r)$ and use your result to find

If $f(r)=r(r+1)(r+2)(r+3)$, simplify $f(r+1)-f(r)$ and use your result to find $$\sum_{r=1}^n r^3$$ I need help with the second part of the question because the book does not give any answers for the ...
0
votes
4answers
43 views

Geometric series weighted by a cosine function

[Edited, corrected notation!] I'm dealing with following series $$ f_d(\omega) := \frac{1-d}{1+d}\sum_{j=-\infty}^{\infty} d^j cos(2\pi j \omega) = \frac{1-d}{1+d}\Big[1 + 2 \sum_{j=1}^{\infty} d^j ...
0
votes
1answer
46 views

Limit of a series of $1$ and $-1.$

If I know that $\lim_{n\to\infty}\sum_{i=1}^n y_i > 0$, where $y_i$ can take the value of $-1$ or $1$, can I assume that the series $\rightarrow \infty$? In other words, if I know the limit is ...
2
votes
1answer
34 views

On the convergence of particular sequences

I'm reading Knopp's book on infinite series. Chapter $2$ which ends with the main results on convergent sequences (not series), suggests the following excercises (the author assumes known the simplest ...
1
vote
2answers
38 views

Find general term of recursive sequences $ x_{n+1}=\frac{1}{2-x_n}, x_1=1/2,$

Please help to solve: $ x_{n+1}=\frac{1}{2-x_n}, x_1=1/2,$ $x_{n+1}= \frac{2}{3-x_n}, x_1=1/2$ I know answers, but can't figure out the solution. The first one is obvious if you calculate first 3-5 ...
7
votes
2answers
124 views

On proving that $\sum\limits_{n=1}^\infty \frac{n^{13}}{e^{2\pi n}-1}=\frac 1{24}$

Ramanujan found the following formula: $$\large \sum_{n=1}^\infty \frac{n^{13}}{e^{2\pi n}-1}=\frac 1{24}$$ I let $e^{2\pi n}-1=\left(e^{\pi n}+1\right)\left(e^{\pi n}-1\right)$ to try partial ...
0
votes
0answers
31 views

Formula for $a_n$ if $a_n=(b(n-1)+c)a_{n-1}+(d(n-2)+e)a_{n-2}$

What is the general formula of a term $a_n$ of a sequence of real numbers $\{a_n\}_{n=0}^\infty$ satisfying $$a_n=(b(n-1)+c)a_{n-1}+(d(n-2)+e)a_{n-2}, \quad n\geq 2,$$ where $b,c,d$ and $e$ are some ...
1
vote
2answers
78 views

On the conjecture that $1-\frac 13 + \frac 16 +\frac 1{10} -\frac 1{15}+\cdots = 1\frac 19$

Conjecture: $$1-\frac 13 + \frac 16 +\frac 1{10} -\frac 1{15}+\cdots = 1\frac 19$$ where the pattern of the signs is $+,-,+,+,-,+,+,+,-,\cdots$ and the denominators are the triangular numbers. ...
3
votes
1answer
114 views

Can anyone help me find the next term for this sequence: 1, 2, 5, 8, 13, 17, 23, 30, 39?

I generated this sequence while working on this problem. Say we have n people trying to pee at n urinals. The urinals and people are all at lattice points (people at (0,0),(1,0),(2,0),...,(n-1,0) and ...
3
votes
3answers
99 views

Find $\lim\limits_{x\rightarrow 0^+}\frac{1}{\ln x}\sum\limits_{n=1}^{\infty}\frac{x}{(1+x)^n-(1-x)^n}$

Find $$\lim\limits_{x\rightarrow 0^+}\dfrac{1}{\ln x}\sum_{n=1}^{\infty}\dfrac{x}{(1+x)^n-(1-x)^n}.$$ Consider $$f(x):=(1+x)^n,$$ By Lagrange MVT, we can obtain $$\frac{2x}{f(x)-f(-x)}=\frac{1}{f'(\...
-3
votes
0answers
15 views

Equating coefficient [closed]

If $\sum\limits_{n = 1}^\infty {{P_n}} \left( t \right){z^n} = z\sum\limits_{n = - \infty }^\infty {{{\left( {\beta z} \right)}^n}} $,Equating the coefficient of ${z^n}$on both sides for n=1,2,3…$\...
-10
votes
0answers
29 views

I want to know what is the sum of this series? [closed]

I want to know what is the sum of $ \sum_{n=1}^{\infty} \frac{1}{n 4^{n}} x^{n} $? Thanks guys!

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