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

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0
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0answers
34 views

sum and infinity

If you have the sums $ (1+2+..+n) + (1+2+3+..+n-1)+ (1+2+3+..+n-2)+(1+2+3+..+n-3)+...+(1+2+3)+(1+2)+1$for large enough $n$ $$\frac {n^3}{3!} \approx (1+2+..+n) + (1+2+3+..+n-1)+ ...
4
votes
1answer
39 views

Prove a sequence converges using sub-sequences

Let there be a sequence $a_n$ The following sub-sequences converge: $a_{n^3},a^3_{2n+3}-a^3_{2n+4},a^2_{2n+3}-a^2_{2n+4},a_{2n+15}$ Prove: $a_n$ converges I think it has something to ...
1
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0answers
14 views

An integral involving the Gamma function

By utilizing the results of two previously asked questions, Series involving Laguerre polynomials and Integral of binomial coefficients, what is a resulting value of the integral \begin{align} ...
1
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0answers
10 views

limiting distribution of a function of joint normals

Let $Z_n=(X_{1,n},X_{2,n})\sim N(\mu,\Sigma_n)$ where $\mu=(0,0)'$ and $$\Sigma_n=\begin{bmatrix}a^2+\frac1n & ab \\ab & b^2+\frac1n\end{bmatrix}$$ Then where does ...
4
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0answers
27 views

A sequence avoiding 3-term power progressions

Rankin1 studied sequences of integers that avoid 3-term geometric progressions, $(a, a c, a c^2)$, e.g., $$\{1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 19, \ldots \} \;$$ So, $18$ is excluded ...
4
votes
7answers
767 views

After switching a lamp on and off infinitely many times in one minute, is it on or off?

So we have a lamp. It's switched on. let's represent its state of being switched on with associating it with $1$ and being off with $-1$. after half a minute passes, you turn it off, after another ...
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2answers
41 views

Determine the digit in a consecutive sequence of numbers

All positive integers are written in order, one after another $$1234567891011121314151617...$$ Which digits appears in the 206 787th position?
5
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3answers
61 views

Convergent $x_n,y_n$ and $x_n^{y_n}$ diverges

Let $x_n, y_n > 0$. I need an example of convergent sequences $x_n,y_n$ such that $x_n^{y_n}$ diverges. Could you help me?
3
votes
1answer
38 views

$f_n$ converges uniformly to f but $f^2_n$ fails to converge uniformly to $f^2$

Consider functional sequence $f_{n}$ which is differentiable on $\left(a,b\right)$. Find an example of $f_n$ converging uniformly to f on $\left(a,b\right)$ such that $f^2_n$ fail to converge ...
1
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1answer
21 views

What is this 2-D array of numbers called?

Define $c(n,r)$ ($n\in\Bbb N;r\in\Bbb Z$) by setting $c(0,-1)=-1$, $c(0,0)=1$, and $c(0,r)=0$ otherwise, with all further $c(n, r)$ given recursively by $$c(n+1,r)=rc(n,r-1)-(r+1)c(n,r)$$(rather in ...
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3answers
33 views

Problem about $\sum_{n=1}^{\infty}n\exp\left(-x\sqrt{n}\right)$

Consider $$\sum_{n=1}^{\infty}n\exp\left(-x\sqrt{n}\right)$$ A. sum of the series is bounded on its set of convergence B. sum of the series is continuous on its set of convergence The correct ...
0
votes
1answer
17 views

Uniform convergence on singleton

First, recall the definition of uniform convergence: Consider functions $f_{n}:A\rightarrow\mathbb{R}$. The sequence of functions $f_{n}$ converges uniformly on set A to limit function f if ...
0
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0answers
28 views

Boundedness of the function

Let $x\in(0,1)$ and $S_{n-1}=\sum\limits_{k=0}^{n-1}x^k$. Then define $f$ as the following :$$f(x)=\sum_{n=1}^{\infty}x^n\left(\frac{n}{S_{n-1}}-1\right)$$ I need to show that ...
3
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2answers
52 views

$\sum_{n=1}^{\infty} \frac{1}{n+1!} \prod_{k=1}^{n} f(k)$ Prove the divergence of a series [on hold]

How can I prove the divergence of the series $$\sum_{n=1}^{\infty} \left(\frac{1}{(n+1)!} \prod_{k=1}^{n} f(k)\right) $$ if $f:\mathbb{N} \rightarrow \mathbb{N}$ is injective? $ $
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2answers
28 views

Show $\lim_{m \rightarrow \infty} \frac{1}{m^2+1} = 0$ using the $\varepsilon$-$N$ definition of convergence

Show that $$\lim_{m \rightarrow \infty} \frac{1}{m^2+1} = 0$$ using the $\varepsilon$-$N$ definition of convergence.
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0answers
32 views

Interpretations of divergent series

I have already bumped into this question and found it not quite useful for my own purposes. The usual way the sum of the infinite series is taught and defined in 1st year undergrad course is that its ...
0
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2answers
63 views

Summation of special series

Does anybody know how to evaluate $$\sum_{i=2}^n(i^2)\cdot{i\choose2}$$ How about the general case of $(i^k)*{i\choose2}$? A nice formula would be great!
0
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2answers
29 views

Summation of “products”

Ok, I am pretty sure I won't get an answer because the question is somewhat hard, and I have done research to no prevail but how can I find the summation of ab if I know the summation of a and the ...
1
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1answer
36 views

Prove that $\prod_1^{\infty}(1-a_i) > 0$ iff $\sum_1^{\infty}{a_i} < \infty$

Suppose $\{a_i\}_1^{\infty} \subset (0,1)$ a) $\prod_1^{\infty}(1-a_i) > 0$ iff $\sum_1^{\infty}{a_i} < \infty$ b) Given $\beta \in (0,1)$, exhibit a sequence $\{a_i\}$ such that ...
5
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0answers
64 views

Is this a known series?

Consider the following series: $$\sum_{p\in\mathcal{P}}\frac{1}{p^p}$$ where $\mathcal{P}$ is the set of all prime numbers: $\mathcal{P}=\{2,3,5,7,11,13,\ldots\}$. My question is: Is this a ...
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4answers
52 views

How do I determine $r$ in this geometric series $a+ar+ar^2+\cdots$? [on hold]

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 .
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2answers
57 views

Evaluate the sum below [on hold]

Evaluate the following sum $$1*1!+2*2!+3*3!+....+1000*1000!$$ any help guys?
4
votes
5answers
169 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
49 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: $$\left\{\frac{2(n+1)}{2(n+1)-1}\right\} - \left\{\frac{2n}{2n-1}\right\}$$ ...
3
votes
5answers
592 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
62 views

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

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
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1answer
15 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
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4answers
56 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?
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0answers
15 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
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1answer
36 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
207 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
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2answers
92 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
319 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
21 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
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2answers
62 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
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0answers
14 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 ...
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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
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4answers
121 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 ...
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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
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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 = ...
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votes
1answer
27 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
114 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
41 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,.......$
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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 ...