For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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0
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1answer
45 views

If $z_n \to a$, then $\frac 1n \sum_{k=1}^n z_k \to a$ [duplicate]

If $\lim_{n \to \infty} z_n = a$, prove that $\lim_{n \to \infty} \frac 1 n \sum_{k = 1}^n z_k = a$. Here is my attempt: $$\left|\lim_{n \to \infty} z_n - a\right| \implies |z_n - a| < ...
1
vote
3answers
56 views

Show that the sequence $a_0 = 1$, $a_{n+1 }= \sqrt{2+a_n}$ is monotonically increasing

Given the sequence $a_0 = 1$, $a_{n+1}= \sqrt{2+a_n}$, how can I show that it is monotonically increasing? I need it to show, that the sequence converges. I already proved boundedness but I can't ...
1
vote
1answer
35 views

random variables and infinite sequence

If $\{x_n\}$ is infinite sequence of random variables and all $\{x_n\}$ values are positive and real numbers then the series $\sum_{n=1}^\infty x_n$ is random variable too Anyone can guide me for ...
4
votes
4answers
445 views

How to find the sum of the infinite series whose general term is not easy to visualize

I am to find out the sum of infinite series:- $$\frac{1}{6}+\frac{5}{6\cdot12}+\frac{5\cdot8}{6\cdot12\cdot18}+\frac{5\cdot8\cdot11}{6\cdot12\cdot18\cdot24}+...............$$ I can not figure out the ...
1
vote
2answers
31 views

For with $j$ and $r$ does $\sum i^{-2r} \lvert \log (2i) \rvert^{-j}$ converge?

I am trying to figure out for which values $j$ and $r$ does the series $$\sum_{i=2}^{\infty} \frac{1}{i^{2r}|\log(2i)|^{j}}$$ converges. I have a feeling that given the exponents ...
0
votes
1answer
9 views

How can one calculate $U_n$ in function of $n$

Given $U_n$ numerical sequence such that : $U_0=-\frac{1}{2}$ $U_{n+1}=\frac{U_n}{3-2U_n}$ The problem is how to calculate $U_n$ in function of $n$ . In this exercise its given the numerical ...
5
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2answers
113 views

How to calculate $\sum_{n \in P}\frac{1}{n^2}, P=\{n \in \mathbb{N}: \exists (a,b) \in\ \mathbb{N^+} \times \mathbb{N^+} \mbox{ with } a^2+b^2=n^2\}$

How I can evaluate $$\sum_{n \in P}\frac{1}{n^2} \quad \quad P=\{n \in \mathbb{N^+}: \exists (a,b) \in\ \mathbb{N^+} \times \mathbb{N^+} \mbox{ with } a^2+b^2=n^2\}$$ It's clearly convergent. I ...
1
vote
2answers
79 views

How to calculate the probability of the next value in a Random Sequence

Assuming that $X$ is the next value in a finite length random sequence, e.g. $$ abbcccaaccbacccabbacababaccccbacabcbacacX $$what would be the $2$-letter most likely to correspond to the value of $X$ ...
-1
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2answers
51 views

Does $\sum_{i=0}^\infty\frac{i^2}{(3+\frac{1}{i})^i}$ converge?

$\sum_{i=0}^\infty\frac{i^2}{(3+\frac{1}{i})^i}$ I thought the only way I could solve this is by comparison after changing the 3 to a 1 which means it would converge. I tried ratio test and thought ...
-4
votes
4answers
64 views

Is this series convergent or divergent $\sum_{n=1}^\infty{\dfrac{\arctan{n}}{n^{2}+1}} $? [closed]

I have problem with convergence $$\sum_{n=1}^\infty{\dfrac{\arctan{n}}{n^{2}+1}} $$ I should use integral test for convergence. Thanks for help.
0
votes
1answer
17 views

Why does $(\frac{n}{n+1})^n{\rightarrow}_{n\rightarrow\infty}\frac{1}{e}$ implies $\frac{2(2n+1)}{n+1}(\frac{n}{n+1})^n|z^2|$ converges?

I want to determine the radius of convergence of the power series $$\sum \frac{(2n)!}{n!n^n}z^{2n}$$ I'm trying to use d'Alembert criterion: Let be $z\in \mathbb{C},\forall c\in N, ...
-2
votes
4answers
80 views

Is this series convergent or divergent $\sum_{n=1}^\infty{\frac{(2n)!}{n^{2n}}}$? [closed]

I have problem with $$\sum_{n=1}^\infty{\dfrac{(2n)!}{n^{2n}}} $$ I try to use Cauchy Condensation Test, but unsuccessfully. Any suggestions? Thanks for any help.
0
votes
1answer
23 views

How to find the decreasing value of a pattern [closed]

I have created an image that will help the understanding of the problem: The number of circles can vary, what I need to find is the constant decreasing value...
1
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3answers
58 views

State whether the following series converges or diverges $\sum\limits_{n=0}^\infty{7^n - 2^n\over(2n)!}$ [closed]

$\sum\limits_{n=0}^\infty{7^n - 2^n\over(2n)!}$ Thanks in advance for any help you are able to provide.
1
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0answers
49 views

Probability of having n more 0's than 1's in a subsequence

We have a number K>2 and a sequence of m binary values $$b_1, ... b_m $$ Given a subsequence $$b_i, ... b_j, i>j $$ we define: W(i,j) is the number of 1's in the sequence L(i,j) is the number ...
3
votes
0answers
31 views

Omega-limit set consists of one point. Does this mean the orbit tends to this point as $t$ grows?

The $\omega$-limit set is defined in this wikipedia article. My question is: If we have an orbit $x_t$ and the $\omega$-limit set of this orbit contains one element $w^*$, does this imply ...
1
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0answers
54 views

Application of algorithm to sequence

I got this puzzle from a professor. Suppose that we have a sequence $a_1$, $a_2$, .., $a_{200}$ $= 1, -1, 0, ..., 0$ where we have $198$ zeros. Now, we perform this algorithm: we replace $a_i$ with ...
9
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1answer
88 views

Sequence related to solutions of the equation $x^x=c$

A couple years ago I remember repeatedly pressing $\sqrt{1+ans}$ into my calculator to be astonished that my calculator gives me an answer approaching the golden ratio. I was astonished, and dug ...
3
votes
2answers
66 views

Testing convergence of the series $\sum \frac{2n-1}{n(n+1)(n+2)}$

I have a series as follows:- $$\sum \frac{2n-1}{n(n+1)(n+2)}$$ I am to test the convergence of this series for natural numbers $n>0$ My approach :- $$\frac{1}{n(n+1)(n+2)}<\frac{1}{n^3} ...
1
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4answers
53 views

Testing the convergence of cube root of some function of n

I have to test the convergence of the following series:- $$\sum_{n=1}^\infty\sqrt[3]{n^3+1}-n$$ My approach is as follows :- $$n^3+1>1=\sqrt[3]{n^3+1}>1=\sqrt[3]{n^3+1}-n>1-n$$ Now since$\sum ...
0
votes
1answer
57 views

Long form of an arithmetic sequence formula

I've been studying arithmetic sequences and am finding that I can do the formulas, but can't truly understand until I can do a long-form version of the formula. Let's take the below: $a$5 = 2+2(5-1) ...
1
vote
1answer
25 views

Find if the recursive sequence is limited and monototonic.

I'm back with another question about sequences which we at the moment we are tought at school. Let $a_{1} = 5$ and recursive formula $a_{n+1} = 5\sqrt{a_{n}-1} -3$ . I think I did it right for ...
4
votes
4answers
138 views

Summing $\sum_ {k=0}^{\infty} \frac{k^3}{3^k}$

How do I find $\sum_ {k=0}^{\infty} \frac{k^3}{3^k}$ . I tried like derivative,like I did in other examples,but in this example that doesn't work... Can somebody help?
1
vote
2answers
52 views

Proof of an infinite sum using Fourier Series

I was revising for my calculus exam and I came across a question that asked to find the Fourier Series of $f(x)=1+x$, on $-1<x<1$, which I did. Which I found to be: $$f(x) = ...
8
votes
1answer
112 views

Near-integer solutions to $y=x(1+2\sqrt{2})$ given by sequence - why?

EDIT: I've asked the same basic question in its more progressed state. If that one gets answered, I'll probably accept the answer given below (although I'm uncertain of whether or not this is the ...
0
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4answers
46 views

Find is sequence is limited and if it's monotonic.

So I have this sequence : $\frac{2n^2 +1}{n^2 +1}$ for every natural number $n>0$. I managed to prove the the lower limit is number $1$ but I cannot figure it out how do I prove that the upper ...
3
votes
1answer
71 views

Can a subsequence of a convergence sum converge anywhere?

This question seems basic but I couldn't find an answer : Let $a_n$ be a decreasing sequence of positive real numbers such that the sum $$\sum_{i=1}^{\infty} a_i$$ converges to $a$ . Then is it ...
0
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0answers
27 views

Inequality issues [duplicate]

hope you're enjoying your holidays. I've been going over my notes, and I found this inequality and I have no clue where it comes from. $\lim_\limits{N \to \infty} \ \inf_\limits{n\ge N} \ ...
1
vote
2answers
71 views

For what values of $\alpha$ does the series $\sum \frac{k^{-\alpha}}{1+\alpha^{-k}}$ converge?

I don't think the series: $$\sum \frac{k^{-\alpha}}{1+\alpha^{-k}}$$ converges for $\alpha \leq 0$ since the terms don't even converge to zero for those values of $\alpha$. I have tried to do a ...
3
votes
1answer
106 views

Showing Convergence of Positive Series

Problem. Let $\{A_{\vec{k}}\}$ be a sequence of real numbers indexed by vectors $\vec{k}=(k_{1},\ldots,k_{n})\in\mathbb{N}$. Let $\{r_{\vec{k}}\}$ be a sequence of positive real numbers such that ...
4
votes
3answers
98 views

Does $\sum_{n=1}^\infty \left(e^{\frac{1}{n}} - 1\right)$ converge?

Evaluate $\sum_{n=1}^\infty \left(e^{\frac{1}{n}} - 1\right)$ $ \lim e^{\frac{1}{n}} = 1$ $\sum_{n=1}^\infty \left(e^{\frac{1}{n}} - 1\right) > \sum_{n=1}^\infty(-1) = -\infty$ I'm not sure how ...
3
votes
1answer
24 views

How can one show that : $|u_{n+1}-\sqrt{2}|\le\frac{1}{4}|u_n-\sqrt{2}|$

$U_n$ numerical sequence such that : ( For all natural numbers $n$ ) $U_{n+1}=1+\dfrac{1}{1+U_n}$ and $U_0=1$ How can one show that : $|U_{n+1}-\sqrt{2}|\le\frac{1}{4}|U_n-\sqrt{2}|$ I arrived to ...
0
votes
0answers
31 views

$X_n$ sequence of independent random variables, with distribution functions: $F_n(x)=\frac{n^x-1}{n-1}, 0\leq x \leq 1, n=1,2,3..$

$X_n$ sequence of independent random variables, with distribution functions: $$F_n(x)=\frac{n^x-1}{n-1}, 0\leq x \leq 1, n=1,2,3..$$ Question convengence. I do not know how to do convergence in ...
3
votes
3answers
48 views

Series converge or diverge : $\sum^\infty_{n=1}n(1+n^2)^p$ , $p\in\mathbb{R}$?

$\sum^\infty_{n=1}n(1+n^2)^p$ , $p\in\mathbb{R}$ I tried to compare it with other known sequencies but i couldn't find the right one. I also tried to solve it using Mathematical Induction (for ...
5
votes
3answers
119 views

Obtaining Fourier series of function without calculating the Fourier coefficients

In this question in one of the answers it's shown how to get from $$f\left ( x \right )=\sum_{n=1}^{\infty}\frac{\sin\left ( nx \right )}{10^{n}}$$ to $$f\left ( x \right )=\frac{10 \sin ...
2
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1answer
97 views

Calculating the value of $\sum^\infty_{k=1}\frac{1}{k(9k^2-1)}$

My first thought is to split it up into: $$\sum^\infty_{k=1}\;\frac{3}{2(3k-1)} + \frac{3}{2(3k+1)} - \frac{1}{k}$$ This is starting to look vaguely like some sort of rearrangement of the ...
5
votes
1answer
95 views

Functions on real line which preserves dfferent modes of convergence and preserves divergence of real infinite series

From this question The set of functions which map convergent series to convergent series , it is known that the set of functions on real line which maps convergent series to convergent series is ...
4
votes
1answer
53 views

Explanation of formula for integer sequence with integers being repeated according to polynom

I have two questions concerning OEIS sequence A056556: $m$ is repeated $\frac{1}{2}(m+1)(m+2)$ times: $$0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, ...
11
votes
4answers
1k views

Is collapsing considered a legitimate proof?

For example if I want to prove that $2^n - 1 = 1 + 2 + 4 + 8 +...+ 2^{n-1}$ I can obviously use induction and that is accepted. But I can also collapse it like: To Prove $2^n = S(n)$: $S(n) = (1 + ...
1
vote
1answer
103 views

An optimal sequence of length 13

I'm looking for an optimal (or much better than I have now) increasing sequence $t_1, t_2, .., t_{13}: t_i \in N, 0 \le t_i<t_{i+1}$ where $3t_{12} + 4t_{13}$ is minimized, subject to the ...
5
votes
1answer
69 views

Proving $\pi=2\sum_{n=0}^{\infty} \arctan \frac{1}{F_{2n+1}}$

How to prove that $$\pi=2\sum_{n=0}^{\infty} \arctan \frac{1}{F_{2n+1}}$$ Where $F_{n}$ is the Fibonacci Number.
1
vote
1answer
77 views

Behavior of interesting sequence

I got this interesting sequence from a friend and I wish to know more about its behavior. I have a sequence of 10, -10, and 198 zeroes. Suppose we, for every number in the sequence, replace it with ...
11
votes
3answers
210 views

Conjecture $\sum_{m=1}^\infty\frac{y_{n+1,m}y_{n,k}}{[y_{n+1,m}-y_{n,k}]^3}\overset{?}=\frac{n+1}{8}$, where $y_{n,k}=(\text{BesselJZero[n,k]})^2$

While solving a quantum mechanics problem using perturbation theory I encountered the following sum $$ S_{0,1}=\sum_{m=1}^\infty\frac{y_{1,m}y_{0,1}}{[y_{1,m}-y_{0,1}]^3}, $$ where ...
4
votes
1answer
40 views

Bounded measurable function $[0, 1] \to \mathbb{R}$ agrees with function of Baire class $2$ outside set of measure zero? Baire class $1$?

We say $f_n \to f$ monotonely on $[0, 1]$ if $f_n \to f$ pointwise and $f_1 \le f_2 \le \dots$ or $f_1 \ge f_2 \ge \dots$. A function $f$ is of Baire class $0$ if $f$ is continuous, and of Baire class ...
3
votes
1answer
65 views

Finite sum: $\sum_{i=1}^n\prod_{j=1,j\neq i}^n\frac{\alpha+z_i-z_j}{z_i-z_j}=n$ [closed]

Any ideas of how to prove: $\sum_{i=1}^n\prod_{j=1,j\neq i}^n\frac{\alpha+z_i-z_j}{z_i-z_j}=n$ where $\alpha$ is a constant. Assume $z_i\neq z_j$.
2
votes
2answers
56 views

Problems with finding limit of $\frac{1}{e^n/n^n - 2(n!)/n^n} $

I have the sequence $$\dfrac{1}{e^n/n^n - 2(n!)/n^n}$$ and by looking at the sequence for $n$ going towards infinity I concluded that the sequence diverges and that it's limit is $ -\infty$. I got to ...
3
votes
1answer
53 views

Example of an infinite sum of functions $f_n(x)$ that converges to $x$, is there a typo in my book?

I have a book that says the following: Let $f_1(x), f_2(x), \dotsc$ a sequence of bounded functions with $f_1(x) + f_2(x) + \dotsb = x$, for example $$ f_1(x) = \frac{\sin x}{x}, \; f_n(x) = ...
4
votes
2answers
106 views

Finding the infinite series: $\sum_{m=0}^\infty\sum_{n=0}^\infty\frac{m!\:n!}{(m+n+2)!}$

Evaluating $$\sum_{m=0}^\infty \sum_{n=0}^\infty\frac{m!n!}{(m+n+2)!}$$ involving binomial coefficients. My attempt: $$\frac{1}{(m+1)(n+1)}\sum_{m=0}^\infty ...
0
votes
1answer
46 views

When is $\frac{\sqrt{n}+(-1)^n}{\sqrt{n+a}}$ defined?

When is $$u_n=\frac{\sqrt{n}+(-1)^n}{\sqrt{n+a}}$$ defined? I understand that it must be defined iff: ($n+a>0$ and $\sqrt{n}+(-1)^n>0$) But then the correction was: ($n>-a$ ...
0
votes
4answers
50 views

Let $f(x)=\sum_{n=0}^{\infty}a_nx^n$ be convergent on $(-R,R),$ with $R>0.$ Prove that, if $f(x)=0$, $a_n=0$.

I'm stuck on a solution that our teacher gave to us. This is the exercise: Let $f(x)=\sum_{n=0}^{\infty}a_nx^n$ be convergent on $(-R,R),$ with $R>0.$ Suppose that $f(x)=0$ for all ...