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

learn more… | top users | synonyms (5)

1
vote
1answer
33 views

Showing a set is not norm bounded

Consider the set $K = \{x(n) : x(n) \in \ell^p, \sum |x(n)| < 1\}$ $(0 < p < 1)$. I have shown that this set is weakly bounded, but I am now asked to show it is not originally bounded. where ...
-1
votes
2answers
59 views

How can I solve the following differential equation without use power series [on hold]

Let we have the following differential equation $$y''-xy'=e^{-x}$$ how can I solve this differential equation without use power series
0
votes
1answer
21 views

Fastest growth rate of sequence?

Sorry about the weird page formatting. So somehow this will use linear algebra to be solved but I'm a bit confused on what to do. I see that (0,0) will work. But c can't be 0. I also see that ...
2
votes
3answers
54 views

If $x_n\cdot y_n\to 0$ then $x_n \to 0$ or $y_n \to 0 $

Assume $$\lim_{n\to \infty}{x_ny_n}=0$$ then $$\lim_{n\to \infty}{x_n}=0$$ or $$\lim_{n\to \infty}{y_n}=0$$ I couldnt find two series who disprove that.. just hint please thanks
4
votes
1answer
60 views

Can we determine whether $f\in L^{p}$ or not ; if we know $\hat{f}$

Let $a_{n}:=\frac{1}{n}$ for all $n\in \mathbb Z\setminus \{0\}$ and $a_{0}= c$ where $c$ is some constant. Clearly, $a_{n}\in \ell^{2}(\mathbb Z)$, that is, $\sum_{n\in \mathbb Z} |a_{n}|^{2}< ...
2
votes
2answers
54 views

Calculate $\sup\limits_{x\in(0,+\infty)}\frac{x^2 e^{-n/x}}{n^2+x^2}$

Calculate $$\sup_{x\in(0,+\infty)}\frac{x^2 e^{-n/x}}{n^2+x^2}$$ The derivative is $$\frac{n e^{-n/x} (n+x)^2}{(n^2+x^2)^2}\geq 0$$ then $$\sup_{x\in(0,+\infty)}\frac{x^2 ...
0
votes
1answer
81 views

Question on finding formula for a Sequences

Hello would appreciate any help, the sequence is given below $y_0 = 1,\,y_1= 2y_0+1 = 2+1 = 3,\,y_2= 2y_1+1 = 2^2 + 2 + 1 = 7,\,y_3= 2y_2 +1 = 2^3 +2^2 +2 +1 = 15$ The question is What is the ...
0
votes
0answers
14 views

How do I 'artificially' construct a closed form for a given sequence?

I've always been a bit iffy about those "Find the next number in the sequence A, B, C, D, ???" problems. Given any finite sequence of numbers, there will (probably?) be an infinite number of general ...
1
vote
1answer
38 views

$\sum \limits_{k=1}^{\infty} \frac{1}{k} \cos 2\pi k \theta$ does not converge as $\theta \rightarrow 0?$

We know that the series $H(\theta) := \sum \limits_{k=1}^{\infty} \frac{1}{k} \cos 2\pi k \theta$ is convergent for every $\theta \in (0,1)$ and for $\theta = 0$ the series tends to $+ \infty$. Is it ...
1
vote
2answers
514 views

Sequence has a convergent subsequence in R^n

Suppose A is a closed and bounded subset of R^n. Let {ak} be a sequence in A. Thus, the elements of {ak} are: (a11,a12,...,a1n), (a21,a22,...,a2n), ... ... (ak1,ak2,...,akn), ... We are not sure if ...
0
votes
0answers
26 views

Series Expansion from Polynomial w/ Coefficients [closed]

I have four coefficients to a 4-the order polynomial. Besides having some stroke of luck finding a pattern (that would be difficult considering the coefficient values) what is the best way to approach ...
0
votes
0answers
9 views

Decomposing integer reciprocals into sums or products with easy binary representation

Background information.(unrelated to mathematics; feel free to skip.) I am looking for good-enough approximations for computing divisions between two (not very big) integers, of which the divisor is ...
1
vote
0answers
40 views

series of powers of integer powers

Given two real positive numbers $a,b\in(0,\infty)$ and a series of natural integers $n=1,2,3,\dots$, is there any known formula to apply in order to calculate the series $$s(n)=a^{b^n}?$$ My goal is ...
3
votes
1answer
67 views

Checking convergence of $\sum\frac{\sin nx}{n}$

Consider the sequence $$f_n(x)=\sum_{k=1}^{n}\frac{\sin kx}{k}\quad x\in \mathbb{R}$$ now we have to check convergence of $\{f_n\}$. Now, I used Dirichlet's criterion to show that $f_n$ converges ...
1
vote
3answers
39 views

formula to divide a distance into X parts with incremental step of Y

I have a mathematical problem that is beyond my ability to solve, so I thought I would ask here: I have a distance between two values: say 100 and 10. And I have a step value say 5. My need is that I ...
1
vote
1answer
24 views

Power Series and Functions [closed]

The function $f(x)=2x\ln (1+x)$ is represented as a power series $$ f(x)=\sum_{n=0}^{\infty}c_nx^n. $$ Find $c_2,c_3,c_4,c_5,c_6$ and the radius of convergence of $f$. http://imgur.com/uLG85Jn Can ...
3
votes
2answers
89 views

Let $u_{n+3} = u_n + 2u_{n+1}$ . Show that $p$ divides $u_p$ for all $p$ prime number.

Let $(u_n)$ a sequence such that $u_0 = 3$, $u_1 = 0$, $u_2 = 4$ and $u_{n+3} = u_n + 2u_{n+1}$ Show that $p$ divides $u_p$ for all $p$ prime number. I'm really stuck on this exercise, ...
1
vote
2answers
35 views

Taking the square of a finite series

I was reading something that involved taking the square of a sum $\sum_{i=1}^k(n_i-1)$. The author just presented the result. $$\left(\sum_{i=1}^k(n_i-1)\right)^2 = \sum_{i=1}^k(n_i^2-2n_i)+k + cross ...
2
votes
1answer
47 views

Does $\sum_{n=1}^{\infty}(-1)^n\frac{n}{\sqrt{n^3 + 6}}$ converge?

Test the series for convergence or divergence. $$\sum_{n=1}^{\infty}(-1)^n\frac{n}{\sqrt{n^3 + 6}}$$ Because this is an alternating series, I decided to use the alternating series test. This ...
1
vote
4answers
41 views

$d_n=(1+(2/n))^n$ converge or diverge and find the limit?

I know the answer is $e^2$ and I'd like to use L'Hopital's rule because this is an indeterminate form. Can someone explain how to get there? $$d_n=\left(1+\frac{2}{n}\right)^n$$
1
vote
8answers
281 views

What Rule generates the sequence $8,8,10,12,12,14,16,16$?

What is the rule that generates the following sequence? I can not solve it. 8 8 10 12 12 14 16 16 For example: 1 3 5 7 9 11 Rule: $n+2$
0
votes
1answer
17 views

functions and recursions

The sequence s(k) where k=1,2,3.... satisfies the recursion s(n)=s(n−2)+s(n−3) for n≥4. If s(n) is rewritten in the form s(n)=s(n−1)+S(n0,n1,…) where S(n0,n1,…) is some linear combination of terms ...
2
votes
1answer
29 views

Sum of $\sum_{k=1}^\infty e^{-x} \frac{x^k}{k!} \sum_{j=1}^k \frac{\lambda^j}{j}$

I would like to simplify the following sum if possible: $$ e^{-x} \sum_{k=1}^\infty \frac{x^k}{k!} \sum_{j=1}^k \frac{\lambda^j}{j} $$ where $x \geq 0$ and $\lambda \in[0, 1]$. When $\lambda = 1$ the ...
-2
votes
0answers
28 views

a question about real analysis,I need to know whether these two sequences are equidistributed. [closed]

For each subset $S\subset [0,1]$ write $X_{s}$ for the "indicator function" as $X_{s}(t)=1$ when $t\in S$ and $X_{s}(t)=0 $ when $t\notin S$. a sequence $\{x_{n}\}$ in [0,1] is called ...
1
vote
1answer
25 views

Finding the radius of convergence for a Maclaurin series

I am required to find the radius of convergence for the function $$f(x) = 5x^3 - 6x^2 - 7x + 6$$ by first finding its Maclaurin series. I found the Maclaurin series to be $$ 6 - 7x -6x^2 + ...
-1
votes
0answers
17 views

Moving part of the sequence to fit a function. [closed]

The sequence $s(k)$ where $k=1,2,3....$ satisfies the recursion $s(n)=s(n-2)+s(n-3)$ for $n\geq 4$. If $s(n)$ is rewritten in the form $$s(n)=s(n-1)+S(n_0,n_1,\dots)$$ where $S(n_0,n_1,\dots)$ is ...
3
votes
2answers
74 views

How did Rudin conclude his argument there is no “boundary” between convergent and divergent series?

I lost my baby Rudin book on real analysis book but I recall a pair of results in homework exercises that he seemed to indicate that there is no "boundary" between convergent and divergent series of ...
0
votes
1answer
32 views

Prove/disprove statement about positive sequence which tends to infinity

Let $\{a_n\}$ be a positive sequence. I have to verify the following statements: If $\lim_{n\to\infty}a_n=\infty$ then $\sqrt[n]{a_n}>1+\frac{1}{n}$, for all but a finite number of $n$ If ...
2
votes
2answers
69 views

Terms needed to approximate with given error?

How many terms of this series would one need to add to get an approximation of $\pi$ with error less than $10^{-4}$? $$ 4 - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \cdots $$ So far, I wrote the ...
0
votes
2answers
32 views

Convergence of infinite series from 2 to infinity 1/(x((lnx)^2))

On a recent exam I was asked to test the following series for convergence From $2$ to $\infty$ $\frac{1}{x(lnx)^{2}}$ I blanked on the integral but set up a comparison test, saying that ...
1
vote
1answer
25 views

Closed-form expression of $\frac{1}{N^{n}} \left[ N^{n+1} - \sum_{k=1}^{N} (k-1)^{n} \right] $

Is there a nice closed-form expression for $$\frac{1}{N^{n}} \left[ N^{n+1} - \sum_{k=1}^{N} (k-1)^{n} \right] $$ where $n, N, k \in \mathbb{N}$. I can obtain an approximation for this for large ...
0
votes
1answer
76 views

Estimate or evaluate the sum: $\sum_{k=1}^n \frac{\cos k x}{k}$.

Estimate or evaluate the sum: $$\sum_{k=1}^n \frac{\cos k x}{k}$$ where $x\in\mathbb R$. My approach: $$f(x)=\sum_{k=1}^n \frac{\cos k x}{k}\Rightarrow f'(x)= - \sum_{k=1}^n \sin k x$$ If we use ...
-1
votes
1answer
56 views

Help me find the rule of this number sequence [closed]

The sequence of natural numbers: 2, 4, 11, 36, 147, 778, ..., 3629814 (10th number in the sequence), ... Help me find the rule of this number sequence
0
votes
2answers
20 views

If $z_n \rightarrow 0$, and if $\{c_n\}$ is bounded, then $\{c_n z_n\} \rightarrow 0$

This is my progress: For complex sequences $\{s_n\}, \{t_n\}$ where $\lim_{n \to \infty} s_n = s,\ \lim_{n \to \infty} t_n = t$, it follows that: $$\lim_{n \to \infty} cs_n = cs \text{ for any ...
0
votes
1answer
19 views

single value computation

I was solving a coding problem where given a bunch of numbers, i need to compute step difference till i'm left with only one number. For example numbers are 3, 5, 2, 6, 7 such that my result is ...
23
votes
2answers
950 views

Finding the limit of a sequence with an undesirable $\ln k$

I am trying to compute the limit of this sequence: $$\lim\limits_{n \to \infty} \dfrac{(-1)^nn^2}{n!} \sum\limits_{k=2}^{n}\binom{n}{k}(-1)^kk^{n-1}\ln k$$ I can compute without the $\ln k$ in the ...
1
vote
3answers
190 views

Convergence of series involving the prime numbers

Given the serie $A=\sum\limits_{n=1}^{+\infty}\frac{p_n}{p_{n+1}}$ and $B=\sum\limits_{n=1}^{+\infty}\frac{p_{2n}}{p_{2n+1}}$, where $p_n$ is the sequence where the nth number are the nth prime ...
1
vote
2answers
57 views

Excerise 12.2 from Ross Elementary Analysis

I'm having a little bit of difficulty proving this question: Prove that $ \limsup_{n->\infty} |S_n| = 0 ~~$iff$ ~\lim_{n->\infty}S_n = 0. $ What I have so far: $ (\Leftarrow)$ $ $Suppose $ ...
0
votes
1answer
27 views

Calculating sequence product

I need your help calculating the limit of: $((n+a_{1})(n+a_{2})...(n+a_{p}))^{1/p}-n$ I've tried to multiply by the conjugate but the expression isn't friendly. Also I've tried to decompose the ...
0
votes
1answer
28 views

Finding the sum of many square root values (greatest integer function)?

I am not really sure how to go about this mathematically, but I used my programming skills to find the answer quite quickly: ...
3
votes
1answer
45 views

Riemann-Stieltjes Integrability and Convergent Series

Let $\alpha_{n=1}^{\infty}$ be a sequence of monotonically increasing functions on $[a.b]$ such that the series $\sum_{n=1}^{\infty}\alpha_{n}(a)$ and $\sum_{n=1}^{\infty}\alpha_{n}(b)$ converge. ...
2
votes
0answers
26 views

Proving a result using Riemann-Stieltjes integration

Let $(\alpha_n)_{n=1}^\infty$ be a sequence of monotonically increasing functions on $[a,b]$ such that the series $\sum_{n=1}^\infty \alpha_n(a)$ and $\sum_{n=1}^\infty \alpha_n(b)$ converge. I must ...
1
vote
0answers
51 views

Evaluate the following series using elementary methods. [duplicate]

$\frac{1} {\sqrt 1} + \frac{1} {\sqrt2} + \dots + \frac{1} {\sqrt{ 100}}$ Just bear in mind that I'm going to say the solution to a person of low education. So please provide creative hints or ...
1
vote
2answers
71 views

Finding the interval of convergence of $\sum^{\infty}_{n=0}\frac{(2n)!}{(n!)^2}x^n$

This is part of a question: Use a similar trick to find nice upper and lower bounds for $\frac {(2n)!}{4^n(n!)^2}$, and thus finish finding the interval of convergence of ...
2
votes
1answer
34 views

Are these recursive sequences convergent?

Fix an integer $k > 1$. Suppose $a_1,\ldots,a_k > 0$ and for $n > k$ we define $$a_n = 1/a_{n-1} + 1/a_{n-2} + \ldots + 1/a_{n-k}$$ Are these recursive sequences always convergent for any ...
5
votes
3answers
166 views

Concerning series of positive real numbers whose terms are decreasing and tending to $0$

Let $\{a_n\}$ be a decreasing sequence of positive real numbers such that $\lim_{n \to \infty} a_n=0$ and $\sum_{n=1}^{\infty}a_n= \infty$ ( for eaxmple , like $a_n:=\dfrac 1n$ ), then is it true ...
0
votes
0answers
18 views

Explain the following summation

Suppose $K=(k_{d-1}, \dots\dots,k_{1},k_{0})_{2^w}$ then $KP = \displaystyle\sum_{i=0}^{d-1}(2^{wi}P)k_i = \displaystyle\sum_{j=0}^{2^{w}-1}\bigg(j\displaystyle\sum_{i:k_i=j}2^{wi}P\bigg)$ After ...
7
votes
1answer
185 views

A problem of Ramanujan's interest: closed form of $1 + 2\sum_{n=1}^{\infty} \frac{\cosh(n\theta)}{\cosh(n\pi)} $

I am Brian Diaz, and I am new to the math.stackexchange community. I have been struggling with attempting to find a closed form of the following series: $$ \varphi(\theta) = 1 + 2\sum_{n=1}^{\infty} ...
3
votes
1answer
55 views

Taylor series expansion of the function $f(x)=x \arctan x-0.5 \log(1+x^2)$ about the origin int the region {$|x|\le1$}

Find the Taylor series expansion of the function $\color {green}{f(x)=x \tan^{-1} x-0.5 \log(1+x^2)}$ about the origin int the region {$|x|\le1$} My effort: I know $\displaystyle \log ...
-4
votes
0answers
31 views

If $|r|<1$,$k\in\mathbb{N}$ and $x_n=r^n$ then $n^k\cdot x_n\to 0$ [closed]

Let $|r|<1$, $k\in\mathbb{N}$ and $x_n=r^n$ for all $n$. Show that the sequence $y_n=n^k x_n$ converges to $0$.