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

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0
votes
0answers
21 views

Sum of series involving factorials [closed]

$$ \sum\limits_{j=0}^{[\frac{n}{l}]}(-1)^{slj}\left( \begin{array}{c} n \\ lj \\ \end{array} \right)^s \frac{x^{kj}}{[(a)_{bj}]^s}, $$ where $l,s,n,k,a,b$ are natural numbers and x is ...
-3
votes
1answer
16 views

Sketching the graph of a sequence of functions [closed]

Does anyone know how to sketch graphs of a sequence of functions? Say, I need to sketch $f_n(x)=\frac{x^{2n}}{1+x^{2n}}$ for $x \in [0, 2]$. But I have no idea where to start. Can someone please help? ...
-1
votes
1answer
35 views

How to derive this inequality

I learnt that for a standard normal random variable $Z$ and positive $x$, we have $$\mathbb P (Z > x) \geq \frac{x}{x^2+1} \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} = \frac{1}{x+\frac{1}{x}} ...
0
votes
0answers
42 views

Show that the series does not converge uniformly on $\mathbb{R}$

My Work: Here, according to the given facts $f(0)=0$ and $f$ is strictly increasing. I proved part (a) and (b) but failed to prove (c). I was going to use the definition (actually wanted to show ...
2
votes
1answer
58 views

Is there a divergent series with “largest” terms?

Suppose $a_n >0$ and $\sum_{n=1}^{\infty}a_n$ converges. Define $$r_n = \sum_{k=n}^{\infty}a_k$$ Does $\sum_{n=1}^{\infty}\frac{a_n}{r_n}$ diverge? My thinking is yes. Could someone give ...
2
votes
0answers
28 views

Order of Magnitude

It is well known (see here, for example) that we have $$ \psi\left(\frac{1}{2},T\right)=\sum_{p\leq T}\frac{1}{p}=\log(\log T)+A+O\left(\frac{1}{\log T}\right), $$ where $\psi(\sigma,T)=\sum_{p\leq ...
1
vote
3answers
183 views

Formulae for sequences

Given that for $1^3 + 2^3 + 3^3 + \cdots + n^3 = \frac{n^2(n+1)^2}{4}$ deduce that $(n+1)^3 + (n+2)^3 +\cdots+ (2n)^3 = \frac{n^2(3n+1)(5n+3)}{4}$ So far: the sequence $(n+1)^3 + (n+2)^3 +\cdots+ ...
0
votes
1answer
64 views

Recurrence relation: $c_{k+1}=c_k+\frac{1}{(k+1)!}$

I have no idea how to proceed solving a recurrence relation like this. I know that the terms approach $e$ but beyond that I have no idea. The relation is$$c_{k+1}=c_k+\frac{1}{(k+1)!} \ \ ; \ c_0=1$$ ...
4
votes
2answers
147 views

Evaluating $\sum_{n=0}^{\infty } 2^{-n} \tanh (2^{-n})$

Reading in some tables pages I found $$\sum _{n=0}^{\infty } 2^{-n} \tanh \left(2^{-n}\right)=\tanh (1) \left(1+\coth ^2(1)-\coth (1)\right)$$ I try to split in two sum using the roots of the ...
0
votes
2answers
74 views

Does anyone understand this proof: If $A$ is closed and bounded $\implies A$ is sequentially compact.

This is how it goes, I will highlight the parts in yellow which I don;t understand why it is , or the idea behind it. $A$ is bounded so $(\forall x \in A)(\exists M > 0)(\|x\|<M)$ Let ...
0
votes
1answer
41 views

Geometric sequence, find the common ratio [closed]

Given: Sum of first five terms is $44$, sum of the next five terms is $-11/8$. Find common ratio and the first term of the series. Also, find the sum of infinity.
7
votes
2answers
192 views

Show that the series converges, but not absolutely.

Show that the series converges, but not absolutely. $\sum_{n=1}^{\infty}( $exp$(\frac{(-1)^n}{n})-1)$. My Try: Let $a_n=$exp$(\frac{(-1)^n}{n})-1$. I was going to use alternating series test ...
1
vote
1answer
25 views

Bernoulli-like generating function

What are the coefficients of the series for: $$\frac x{e^x+1}$$ It looks similar to the Bernoulli generating function, but the $+$ sign is throwing me off. I already found the series for its ...
3
votes
2answers
105 views

The converges of $ \sqrt { 2 } +\sqrt { 2-\sqrt { 2 } } +\sqrt { 2-\sqrt { 2+\sqrt { 2 } } } + …=$

I would like to know wheather this series converge or diverge? $\sqrt { 2 } +\sqrt { 2-\sqrt { 2 } } +\sqrt { 2-\sqrt { 2+\sqrt { 2 } } } +\sqrt { 2-\sqrt { 2+\sqrt { 2+\sqrt { 2 } } } } ...
0
votes
1answer
19 views

Possible value of $x$ so that fractions are in simplest form.

Which of the following could be the possible value of $x$ for which each of the fractions is in its simplest form, where $\lfloor{x\rfloor}$ stands for greatest integer less than or equal to ...
32
votes
6answers
914 views

A series involving the harmonic numbers : $\sum_{n=1}^{\infty}\frac{H_n}{n^3}$

Let $H_{n}$ be the nth harmonic number defined by $ H_{n} = \sum_{n=1}^{n} \frac{1}{k}$. I'm interested in knowing how to show that $$\sum_{n=1}^{\infty}\frac{H_n}{n^3}=\frac{\pi^4}{72}.$$ I tried ...
30
votes
3answers
850 views

Generalized Euler sum $\sum_{n=1}^\infty \frac{H_n}{n^q}$

I found the following formula $$\sum_{n=1}^\infty \frac{H_n}{n^q}= \left(1+\frac{q}{2} \right)\zeta(q+1)-\frac{1}{2}\sum_{k=1}^{q-2}\zeta(k+1)\zeta(q-k)$$ and it is cited that Euler proved the ...
2
votes
3answers
153 views

How to prove that $\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}$ without using Fourier Series [duplicate]

Can we prove that $\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}$ without using Fourier series?
3
votes
2answers
70 views

$\int_a^b f(x)g(x)dx = \sum \int_a^b f_n(x)g(x)dx.$

Let $\sum f_n(x) $ be uniformly convergent to $f(x)$ on $[a,b]$ where each $f_n$ is continuous on $[a,b]$. If $g: [a,b] \to \mathbb R$ be integrable on $[a,b]$, then $$\int_a^b f(x)g(x)dx = \sum ...
0
votes
2answers
37 views

convergence of the series for $p>0$ , $\sum n^p\left(\frac{1}{\sqrt{n-1}}-\frac{1}{\sqrt n}\right)$

Discuss the convergence of the series for $p>0$ , $$\sum n^p\left(\frac{1}{\sqrt{n-1}}-\frac{1}{\sqrt n}\right)$$ I tried through ratio test. But it fails. I think it will be by comparison ...
0
votes
1answer
53 views

Integer series will become constant? [closed]

Given a bounded integer sequence $(a_n)_{n\in\mathbb{N}}$, prove or disprove that if the sequence $(a_n + n)_{n\in\mathbb{N}}$ has no repetitive elements: $$ \forall n,m \in \mathbb{N}, \,\,m \neq n ...
0
votes
1answer
19 views

Combinatoric for number of ways to have monotone-increasing sequence

I hope I am using the right term. By monotone-increasing I mean to imply that it is a non-decreasing sequence. So for example a sequence $1, 1, 2, 5, 6, 10, 10, 11$, etc. Anyhow, consider a ...
2
votes
1answer
113 views

Compute this integral

$$ \displaystyle \int_{0}^{\infty}{\frac{{log}^{2}(1-{e}^{-x}){x}^{5}}{{e}^{x}-1} dx} $$ What I have done - $ \displaystyle I(k) = \int_{0}^{\infty}{\frac{{x}^{5}}{{e}^{x}{(1-{e}^{-x})}^{k}}}$ ...
1
vote
1answer
25 views

Applying squeeze theorem to conditionally convergent series

Suppose that the series ∑_n≥1(a_n) converges conditionally. Then by the Riemann Series theorem, for any real number L there exists a rearrangement of a_n(let's call it b_n) that converges to L. For a ...
2
votes
1answer
484 views

Convergence of product of sequences in $\ell^2$

I have the following question: Let $(a_{j})$ be a sequence in $\ell^{2}$ and $(b_{j}^{n})$ be a sequence of sequences in $\ell^{2}$ such that $(a_{j}b_{j}^{n})$ is an $\ell^{2}$ sequence for each ...
4
votes
2answers
205 views

Finding a recurrence for a sum

I am trying to implement the following sum using a programming language: $$\sum_{i=1}^N a^i i^r$$ where $N$, $a$ and $r$ are integers. The problem is, I cannot find a suitable way to do this. ...
11
votes
2answers
234 views

Easier ways to prove $\int_0^1 \frac{\log^2 x-2}{x^x}dx<0$

Prove that $$\int_0^1 \frac{\log^2 x-2}{x^x}dx<0$$ One way to do this is use the idea in the proof of Sophomore's dream. We have $$x^{-x}=\exp(-x\log x)=\sum_{n=0}^\infty\frac{(-1)^nx^n\log^n ...
0
votes
1answer
61 views

Convergence of the series $\sum_{n=0}^\infty \sin(n! \pi m \sin(1))$

In this exercise I was asked to prove the convergence of the following infinite sum: $$\sum_{n=0}^\infty \sin(n! \pi m \sin(1)),$$ where $m$ denotes any integer. I don't have any idea on how to ...
12
votes
5answers
411 views

A closed form for the infinite series $\sum_{n=1}^\infty (-1)^{n+1}\arctan \left( \frac 1 n \right)$

It is known that $$\sum_{n=1}^{\infty} \arctan \left(\frac{1}{n^{2}} \right) = \frac{\pi}{4}-\tan^{-1}\left(\frac{\tanh(\frac{\pi}{\sqrt{2}})}{\tan(\frac{\pi}{\sqrt{2}})}\right). $$ Can we also find ...
1
vote
0answers
28 views

Is this :$\sum_{n=1}^{\infty} \frac{\sigma_{k}(n)}{n!} $ irrational series for every natural number $k$?

Is this: $$\sum_{n=1}^{\infty} \frac{\sigma_{k}(n)}{n!} $$ irrational series for every natural number $k$? Where : $\sigma_{k}(n)=\sigma(\sigma(\sigma(\dots n)))$ is the $k$-th iterate of the sum of ...
1
vote
1answer
30 views

General methods of solving non-linear recurrences

Let's consider a sequence $$ \{a_{n} \} _{n=0}^{\infty}, a_{0}=1, a_{n+1}=\frac{a_{n}}{1+na_{n}}$$. Taking $$b_{n}=\frac{1}{a_{n}},$$ we bring the exact reccurence to the following one: $$b_{0}=1, ...
6
votes
3answers
89 views

Show that $x_n$ is convergent.

My Try: It is clear that $x_n$ is monotonically increasing. If we assume that the sequence converges to $a$ then $\displaystyle a=a+\frac{\sqrt{|a|}}{n^2}$. Hence $a=0$. So, I was going to prove ...
2
votes
1answer
65 views

Convergence of series in a Hilbert Space

I'm hoping for some help on the following question. I haven't gotten very far: Let $\{h_n\}_{n\geq 1}$ be a sequence of vectors in a Hilbert space $H$ with the property that $(h_n-h_m)\perp h_m$ ...
2
votes
2answers
56 views

Convergence of an oscillating recursive sequence

Define the recursive sequence $ q_{n+1} = \dfrac{q_n+2}{q_n+1};\;q_0=1 $ If we knew that $ q_n \to q;\;n\to \infty $ then it's easy to show what follows $ q_{n+1}\left(q_n+1\right) = q_n+2 $ $ ...
2
votes
1answer
29 views

Convergence of a sequence of $2\times 2$ real matrices

My Try: So $a_n$ can be written as a series very similar to the taylor series of sin: $\displaystyle a_n=\sum_{k=0}^n \frac{(-1)^k b_k}{(2k+1)!}$ for some $b_k$ to be determined. But it is very ...
0
votes
1answer
43 views

Evaluating a cube root

How to evaluate $(8.024)^{1/3}$ from $(1+3x)^{1/3}$.I already expand it until $x^3$ but i still can't get the answer. I tried googling for the working using binomial theorem but i failed.
3
votes
2answers
113 views

A curious partitions coincidence $\sum_{n=0}^\infty P(n) q^{n+1}$?

Given the partition function $P(n)$ and let $q_k=e^{-k\pi/5}$. What is the reason why, $$\sum_{n=0}^\infty P(n) q_2^{n+1}\approx\frac{1}{\sqrt{5}}\tag{1}$$ $$\sum_{n=0}^\infty P(n) ...
0
votes
1answer
81 views

How to show that his series converges or diverges using LCT or CT?

$$\sum_{n=1}^{\infty}\left (\sqrt{n^4+1}-n^2\right)$$ The question states that either the limit comparison or comparison test can be used to determine whether the series converge or diverge. I tried ...
-3
votes
1answer
21 views

Convgergence of series with different parametes [closed]

I have this series : $$\sum\limits_{n=1}^{\infty} n^{-a}\log(n)^{-b}$$ For what values of $a$ and $b$ does the series converge?
0
votes
1answer
51 views

Ratio in sequences and series

In a geometric series where the common ratio is $r$ ($r^2$ is not $1$) the sum of the first $13$ terms is three times the sum of the first $6$ terms.How do I find in any order the ratio of the sum of ...
3
votes
4answers
399 views

1+4+10+20+35+…=? [closed]

Is there a finite value to the infinite sum of all the tetrahedral numbers: $$\sum_{n=1}^\infty \frac{n(n+1)(n+2)}{6}.$$ I know it's a divergent series, but I hear that $$ ...
4
votes
1answer
135 views

Closed form for this sum?

$$\displaystyle \sum_{m=0}^{\infty}{\frac{{\left({H}_{m}^{(1)}\right)}^{2} - {H}_{m}^{(2)}}{{(m+1)}^{6}}}$$ where $ \displaystyle {{H}_{k}}^{(r)} = \sum_{i=1}^{k}{\frac{1}{{i}^{r}}} $ I have no ...
2
votes
1answer
35 views

FSR function of the component-wise product, sum, of two LFSR sequences

Let $T_1$, $T_2$ be two $m$-sequences over $\mathbb{F}_q$ of length $q^n-1$, say $T_1 = (\text{Tr}_{q^n | q}(\alpha^i))_{i \geq 0}$, $T_2 = (\text{Tr}_{q^n | q}(\beta^i))_{i \geq 0}$, for some ...
1
vote
2answers
60 views

Does this series converge or diverge and by which test?

$$\sum_{n=1}^\infty (-1)^{n+1} \sin(1/n^3)$$ I tried to apply the divergence test. I know $\lim_{n\to \infty}$ is 0 for $b_n$ but I don't think $b_n$ is decreasing. any ideas on how I can test this ...
0
votes
1answer
15 views

Separability of $l^{p}$ spaces

How can I prove that the space $l^{p}$ equipped with the norm (for $x=(x_{n}) \in l^{p})$: $||x||_{p}=(\displaystyle\sum_{n}|x_{n}|^{p})^{1/p}$ Is a separable space? (i.e. showing that there is a ...
1
vote
2answers
78 views

How to find the limit of a sequence?

Question: If $0 < x < \frac{\pi}{2}$ and $f_k(x) = \tan(x)+\frac{1}{2}\tan(x/2)+ ...+\frac{1}{2^k}\tan(x/2^k)$. In Sigma Notation: $$f_k(x) = \sum_{n=0}^k \frac{1}{2^n}\tan\frac{x}{2^n}$$ ...
0
votes
1answer
38 views

Prove that $T_n$ satisfy $ \sum_{k=0}^{N-1}{T_i(x_k)T_j(x_k)} = \begin{cases} 0 &: i\ne j \\ l\neq 0 &: i=j \end{cases} \,\! $

The Chebyshev polynomials of the first kind satisfy the recurrence relation $$ \begin{cases} T_{n}(x)=2xT_{n-1}(x)-T_{n-2}(x) \qquad n \geq 2 \\ T_{0}(x)=1, \ \ T_{1}(x)=x \\ \end{cases} $$ The ...
5
votes
2answers
107 views

Determine all real $x$ for which the series $\sum\limits_{k=1}^\infty\frac{k^k}{k!}x^k$ converges.

Determine all real $x$ for which the following series converges: $$\sum_{k=1}^\infty\frac{k^k}{k!}x^k.$$ You may use the fact that $$\lim_{k\to\infty}\frac{k!}{\sqrt{2\pi k}(k/e)^k}=1.$$ ...
1
vote
4answers
70 views

Sum of roots of unity, proving that $1+w+w^2…+w^{n-1}=0$ [closed]

If $w$ is a unit square of rank $n$ (meaning $w^n=1$), s.t $w$ is not $1$. Prove that $1+w+w^2.....+w^{n-1}=0$. We're pretty sure that we need to use induction, its easy to prove for $n=2$ but ...
5
votes
1answer
139 views

How can I show that the sequence $x_n^2$ is bounded?

Two real sequences $(x_n)$ and $(y_n)$ are defined by $$x_{n+1}=x_n-(x_ny_n+x_{n+1}y_{n+1}-2)(y_n+y_{n+1})$$ $$y_{n+1}=y_n-(x_ny_n+x_{n+1}y_{n+1}-2)(x_n+x_{n+1})$$ with initial conditions $x_0=1$ and ...