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

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0
votes
2answers
21 views

Find the value of this $2f\left ( \frac{1}{2} \right )$

IF $f(f(x))=1-x$, Find $$2f\left ( \frac{1}{2} \right )=??$$ help guys, I really tried but I couldn't.
-2
votes
4answers
38 views

What is $r$ in this infinite geometric series $a+ar+ar^2+\cdots$?

The geometric series $a+ar+ar^2+\cdots$ has a sum of $12$, and the terms involving odd powers of $r$ have a sum of 5. What is $r$? Thank you for any help .
0
votes
2answers
33 views

Evaluate the sum below

Evaluate the following sum $$1*1!+2*2!+3*3!+....+1000*1000!$$ any help guys?
3
votes
4answers
51 views

Binomial coefficients in Geometric summation

Guys please help me find the sum given below. $$\sum_{k=j}^i\binom{i}{k}\binom{k}{j}\cdot 2^{k-j}$$ (NOTE):The two coefficients are multiplied by 2 power (k-j) I am using the formula: ...
1
vote
2answers
45 views

Show that the sequence $\left\{\frac{2n}{2n-1}\right\}$ is monotone by using $a_{n+1} - a_{n}$

Note: I am looking at the sequence itself, not the sequence of partial sums. Here's my attempt... Setting up: $$\left\{\frac{2(n+1)}{2(n+1)-1}\right\} - \left\{\frac{2n}{2n-1}\right\}$$ ...
4
votes
5answers
193 views

Find the value of this series

what is the value of this series $$\sum_{n=1}^\infty \frac{n^2}{2^n} = \frac{1}{2}+\frac{4}{4}+\frac{9}{8}+\frac{16}{16}+\frac{25}{32}+\cdots$$ I really tried, but I couldn't, help guys?
1
vote
3answers
56 views

About the sum of $\sum_{n=1}^{\infty} \frac 1 {n(n+1)}$

Find the sum of $\displaystyle \sum_{n=1}^{\infty} \frac 1 {n(n+1)}$ So I can see that it's a telescopic sum: $\displaystyle \sum_{n=1}^{\infty} \frac 1 {n}-\frac 1 {n+1}$, but since the sum ...
1
vote
1answer
14 views

Recursive Equation : $X_t=-\sum_{j=1}^{\infty}\phi^{-j}W_{t+j}$

$X_t=\phi X_{t-1}+W_t$ $\Rightarrow X_{t-1}=\frac{1}{\phi} X_{t}-\frac{1}{\phi}W_t$ Where , $|\phi|>1$ . But how does the following recursion relation occur : ...
1
vote
4answers
55 views

Monotonicity and convergence of the sequence $a_n=\sum_{k=1}^{n}\frac{1}{k+n}$

Let we have the following sequence $(a_n)$ such that $$a_n=\sum_{k=1}^n\frac{1}{n+k}$$ How can I prove that $(a_n)$ is increasing bounded sequence, then prove it is convergent and find its limit?
0
votes
0answers
13 views

Convergence of Cauchy Product (is Mertens' theorem strong?)

According to Mertens' theorem, if two series are convergent, with at least one of them being absolutely convergent, then their Cauchy product converges to the product of the two series. The ...
0
votes
1answer
33 views

Prove that:$\sum_{i=1}^n\frac{1}{x_i}-\sum_{i<j}\frac{1}{x_i+x_j}+\sum_{i<j<k}\frac{1}{x_i+x_j+x_k}-\cdots+(-1)^{n-1}\frac{1}{x_1+\ldots+x_n}>0.$?

Is there someone show me how do I prove this , I guess this inequality hold only if $x=0$ . Let $x_1,x_2,\ldots,x_n>0$. Prove that ...
4
votes
2answers
202 views

Recursive sequence of square roots of previous elements

Bruckner, Bruckner, Thompson - Elementary Real Analysis $a_1 = 1$ and $a_{n+1} = \sqrt{a_1+ a_2 + .. + a_n}$ Show that $$\lim_{n \to \infty}\ \frac{a_n}{n} = \frac12$$ I cannot untangle the square ...
6
votes
2answers
88 views

$ \lim_{n\to+\infty} \frac{1\times 3\times \ldots \times (2n+1)}{2\times 4\times \ldots\times 2n}\times\frac{1}{\sqrt{n}}$

Knowing that : $$I_n=\int_0^{\frac{\pi}{2}}\cos^n(t) \, dt$$ $$I_{2n}=\frac{1\times 3\times \ldots \times (2n-1)}{2\times 4\times \ldots\times 2n}\times\dfrac{\pi}{2}\quad \forall n\geq 1$$ ...
0
votes
1answer
23 views

proving convergence of this sequence and calculating limit

So i have this sequence: $a_{n}=1+ \frac{1}{3}\cos1 +\frac{1}{3^2}\cos2+ ... +\frac{1}{3^n}\cos(n) $ I have to prove it is convergent, and then calculate the limit. I'm not totally sure how to find ...
2
votes
1answer
61 views

How to solve this seemingly easy problem?

In short, I need to prove that: $\sin 2nx\not\to-\sin 2x\quad x\ne\frac{k\pi}2,k\in\Bbb Z,n\in\Bbb N\quad \text{as}\quad n\to\infty$ The biggest trouble is that I know little about $x$, not even ...
0
votes
6answers
304 views

calculating 2 sums of series

So I have these two series given. 1: $\displaystyle\sum_{n=1}^{\infty}\frac{\sin{(2n)}}{n(n+1)} $ And I have to show that this sum is $\leq$ 1. 2: ...
0
votes
1answer
19 views

Why is this true - easy question concerning asymptotics of exponential

Suppose $\lambda > 0$ is constant as $t \searrow 0$. In my lecture notes it is written that $\left(1+\sum_{k=1}^{\infty} \frac{(-\lambda t)^k}{k!}\right) \lambda t = \lambda t + o(t)$ and ...
1
vote
2answers
61 views

find $\lim_n\frac{a^\frac1n}{n+1}+\frac{a^\frac 2n}{n+\frac 12}+…+\frac{a^\frac nn}{n+\frac 1n}$ using Riemann integral

Here is the question: prove that $S_n=\frac{a^\frac1n}{n+1}+\frac{a^\frac 2n}{n+\frac 12}+...+\frac{a^\frac nn}{n+\frac 1n}$ is convergent for $a>0$ then find its limit. My attempt: If we accept ...
0
votes
0answers
13 views

Why do we need to worry about convergence in $\mathbb{R^Z}$ if each $\mathbf{e}_i$ are already pairwise linearly independent?

(Note: as pointed out by some users in related questions, the $\mathbb{R^\infty}$ in the link turns out to be $\mathbb{R^Z}$) Once again, a question inspired from reading this An excerpt One ...
-4
votes
2answers
25 views

Prove Sum Approximation Theorem [on hold]

Prove if $S=\sum_{n=0}^{\infty}a_{n}x^{n}$ converges for $|x|<1$, and if $|a_{n+1}|<|a_{n}|$ for $n>N$, then $$|S-\sum_{n=0}^{N}a_{n}x^{n}|<|a_{N+1}x^{N+1}|/(1-|x|)$$ I have already proven ...
2
votes
3answers
67 views

$I_{2n}=\dfrac{1\times 3\times \ldots \times (2n-1)}{2\times 4\times \ldots\times 2n}\times\dfrac{\pi}{2}\quad \forall n\geq 1$

let $$I_n=\int_0^{\frac{\pi}{2}}\cos^n(t) \, dt$$ show that $$I_{2n}=\frac{1\times 3\times \ldots \times (2n-1)}{2\times 4\times \ldots\times 2n}\times\dfrac{\pi}{2}\quad \forall n\geq ...
3
votes
4answers
119 views

Does this series converge, and if so to what value?: $\sum_{n=0}^\infty \left\{\frac{1}{(n+1)^2} \right\}\ln(2n+1)$

I've arrived at this series from a given sequence of terms, but now I'm at a loss as to how to proceed... How does one know which convergence test to use? This isn't a geometric series, so I don't ...
-1
votes
2answers
19 views

Help understanding the sequence of partial sums of a series

I just completed calc 2 and am compiling and rewriting my notes for future reference. One thing I found online (Wikipedia etc) but not really covered in my texts is the concept that a series contains ...
3
votes
3answers
56 views

Strange integral test for convergence in my Analysis Script (proof flawed ?)

Today I was going through my Analysis Script which my Professor used for his course (meaning he often refers to it) and I found a Lemma called Integralcriteria for convergence of Series. I read its ...
1
vote
0answers
26 views

Sum of gamma-ish power series

I'm wondering if there is a nice closed-form expression for the sum $$ \sum_{n=0}^{\infty} n^{-\alpha} x^n, \quad \alpha \in (1,2), \; x \in (0,1) $$ This is a power series with coefficients $a_n = ...
1
vote
0answers
47 views

Showing that $U_n$ and $U_m$ must have primes between them

Given that $$U_n=\underbrace{1\cdots1}_{n\text{ times}}$$ and $n >2$, how can one show that if $m$ and $n$ have primes between them, then $U_n$ and $U_m$ must also have primes between them? ...
-2
votes
1answer
26 views

Evaluate lim through progression

Let $a_n$ be the $n$-th term of an arithmetic progression with the initial term $a_1=2$ and with the common difference $5$. That is, $a_n=2+5(n-1)$. Evaluate ...
0
votes
5answers
110 views

What we get if we add 1/2 infinite times [on hold]

I want to know if this is correct We have this sums: $$S1=1-1+1-1+1-1+1-1+1-1...=\frac12$$ $$S2=1-2+3-4+5-6+7-8...=\frac14$$ $$S3=1+2+3+4+5+6+7+8...=-\frac{1}{12}$$ If we take ...
2
votes
1answer
28 views

Derive a formula for the number of small square base pyramids required to create a bigger pyramid?

To quote from the problem statement: "Pyramids are built using smallest pyramids of "level 1", that are used as building blocks for higher levels. Stacking pyramids of "level 1" to create ...
1
vote
1answer
37 views

Help with a summation+inequality problem.

I need help in solving for all possible x values for the below inequality: (Note: $x \in N)$ $$\sum^x_{k=1}\frac{k^2+k+1}{k(k+1)(k+1)!} \leq \frac{599}{600}$$ I think the series is telescopic; I'm ...
0
votes
1answer
39 views

summation problem [on hold]

what is the result for the following double summation: $\sum\limits_{i \neq j}^{\infty}\alpha^i\alpha^j$ where $ i, j =0,1,2,.......$
-1
votes
1answer
52 views

Find the limit of $A={\{(\dfrac{\theta-1}{\theta}}, \theta)|\theta=1,2,3,\dots\}$

Question: Find the limit of $A={\{(\dfrac{\theta-1}{\theta}}, \theta)|\theta=1,2,3,\dots\}$? Here, $\left(\dfrac{\theta-1}{\theta},\theta\right)$ is a point in $\mathbb R^2$ expressed in polar ...
6
votes
2answers
48 views

Convergence and divergence depending on whether $n$ is odd or even

It is part of one problem I am working on: I want to prove the following conjecture ($x\ne q\pi$ where $q\in\Bbb Q$) $$\sum_{n=1}^{\infty}\frac{\sin^{2k-1}(nx)}{n^{\alpha}}\quad\text{converges ...
3
votes
1answer
43 views

No convergence in Discrete metric - why?

If I have a sequence $(x_n)_{n \in \mathbb{N}} := (\cfrac{1}{n},\cfrac{1}{n})$ in $\mathbb{R}^2$ Then why isn't there convergence with respect to the Discrete metric? for a discrete metric, the ...
1
vote
1answer
19 views

Series Test for Integrability via the Distribition Function

I imagine that the following question has a well known (and perhaps, easily obtainable) answer, but I can't find it by myself nor along the references that I have in mind so far. So, if $f$ is a ...
-5
votes
0answers
37 views

What is the sum of given series? [on hold]

Series is $3+3 \cdot 3 +3 \cdot 6$ up to the $n$-th term. How do we formulate the AP series for this type of problems? What is the sum?
0
votes
2answers
78 views

Summing divergent asymptotic series [on hold]

I found the sine integral si to be $$Si (x)\sim \frac \pi 2+\sum _{n=1}^\infty (-1)^n \left(\frac{(2 n-1)! \sin (x)}{x^{2 n}}+\frac{(2 n-2)! \cos (x)}{x^{2 n-1}}\right)$$ Say I want to find ...
-5
votes
1answer
38 views

What should be next numbers [on hold]

$6\ 6\ 6\ 2\ 5\ 7\ \ \ \ \ \ \ \ \ \ 1\ 2\ 5\ 6\ 6\ 3$ $6\ 9\ 3\ 7\ 7\ 9\ \ \ \ \ \ \ \ \ \ 1\ 7\ 4\ 2\ 0\ 9$ $4\ 2\ 9\ 7$ What would be next 8 numbers
1
vote
5answers
216 views

I need help with a Finite Series

Problem: Find the sum to $n$ terms of \begin{eqnarray*} \frac{1}{1\cdot 2\cdot 3} + \frac{3}{2\cdot 3\cdot 4} + \frac{5}{3\cdot 4\cdot 5} + \frac{7}{4\cdot 5\cdot 6}+\cdots \\ \end{eqnarray*} ...
1
vote
3answers
40 views

Finding an explicit formula for a recursive sequence. [on hold]

How to show that the recurrent formula $$A_n=A_{n-1} + A_{n-2} +4.$$ gives a sequence of the form $f(n)=cr^n+cr^n$? The only way we are allowed to solve it, is with the quadratic formula ...
4
votes
0answers
56 views

Long polynomial expansion with 34 roots

This is a very tricky problem, I just need a few hints. I think the $(-x^{17})$ is also there for a specific trick. In the end if it is $ax^{17}$, I see that $a = 17 - 1 + 1 = 17$. Also, another ...
2
votes
1answer
111 views

Given: $\sum n a_n$ is convergent. To prove: The sequence ${a_n}$ converges

Given: $\sum n a_n$ is convergent. To prove: The sequence ${a_n}$ converges. My proof: Since $\sum n a_n$ is convergent, so $na_n \rightarrow 0,$ i.e., $\frac{a_n}{\frac{1}{n}}\rightarrow 0. $ As ...
2
votes
2answers
34 views

Find Taylor expansion of $f(x)=\ln{1-x^2\over 1+x^2}$, and then find radius of convergence.

Find Taylor expansion of $f(x)=\ln{1-x^2\over 1+x^2}$, and then find radius of convergence. My problem here is the Taylor series. Computing the few first derivative is possible, but I can't seem to ...
0
votes
0answers
23 views

How do I formulate a specific formula for a sequence?

I have three arrays, for instance s = [1:2], j = [1:20] and b = [1:8], and I am trying to build a single row. The problem that I actually have is that I need to find a formula f(s,j,b) such that ...
0
votes
1answer
41 views

suppose n is a natural number , prove equation $x^n+nx-1=0$ exist an unique real positive root $x_n$

suppose n is a natural number prove : equation $x^n+nx-1=0$ exist an unique real positive root $x_n$ ; and when $a>1$,$\sum_{n=1}^{\inf}x^a_n$ converges.
10
votes
2answers
423 views

How long will this take to reach.. kimye?

I found the 1bc29b36f623ba8 twitter account on 4chan last night, it's a user who is posting md5 hashes every 10 minutes in a sequential order, starting at ! and I ...
0
votes
1answer
34 views

How many ways could up to n factors sum up to n

With n = 2, we only have 2 ways of adding integers to produce it: 2 and 1 + 1. With ...
3
votes
0answers
73 views

Calculate in closed form $\sum_{n=1}^{\infty} \frac{\arctan(1/n) H_n}{n}$

Playing with Taylor series is not helpful enough. What else would you try out? $$\sum_{n=1}^{\infty} \frac{\arctan(1/n) H_n}{n}$$ $$\approx 2.1496160413898356727147400526167103602143301206321$$ It's ...
0
votes
1answer
67 views

My answer is coming out wrong in this series.

$$\lim_{n\to\infty}\left( \frac 1{1-n^2}+\frac 2{1-n^2}+\frac 3{1-n^2}+\cdots+\frac n{1-n^2} \right)$$ My answer is coming out to be $-\frac 12$. I factored out the common $\frac 1{1-n²}$, summed up ...
1
vote
1answer
48 views

Lim Sup inequality

Elementary Real Analysis, Thompson Bruckner & Bruckner. For any sequence ${a_n}$, write $s_n = \frac1n\sum_{i=1}^n a_n$ Prove that $\limsup s_n \leq \limsup a_n$ and give an example that the ...