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

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0
votes
1answer
40 views

Graphic proof of an inequality between sequence ratios

I would like to verify my proof for the following claim. Let $b_i$ be a positive decreasing sequence, $j<k$ two integers and $d$ a positive number. Prove that: $$ ...
4
votes
3answers
74 views

Sum of $\sum_{n=1}^{\infty}(-1)^{n+1}\frac{x^{2n-1}}{2n-1}$

I've been working with the series: $$\sum_{n=1}^{\infty}(-1)^{n+1}\frac{x^{2n-1}}{2n-1}$$ From the ratio test it is clear that the series converges for $|x| < 1$, but I'm unable to obtain the sum ...
4
votes
5answers
77 views

Calculate $\lim_{n\to\infty} (n - \sqrt {{n^2} - n} )$

Calculate limit: $$\lim_{n\to\infty} (n - \sqrt {{n^2} - n})$$ My try: $$\lim_{n\to\infty} (n - \sqrt {{n^2} - n} ) = \lim_{n\to\infty} \left(n - \sqrt {{n^2}(1 - \frac{1}{n}} )\right) = ...
1
vote
1answer
50 views

Convergence of $\sum_{n=1}^{\infty}(-1)^n\frac{x^n+1}{n}$

I have trouble with the following sum: $$S=\sum_{n=1}^{\infty}(-1)^n\frac{x^n+1}{n}$$ Since $a_n=\frac{2}{n}$ decreases monotonically and tends to $0$, it converges by Liebniz criterion. Then, by ...
1
vote
4answers
67 views

How to prove that the following sequence will never contains number greater than 3

You may now the following sequence: 1 11 21 1211 111221 312211 13112221 Explanation of the sequence (I've put an hint just for the ones who want to search a bit ...
0
votes
0answers
21 views

Convergence of a sequence using Banach contraction principle: $x_{n+1}=\frac{1}{x_n+a},x_1>0,a \in \mathbb{R},n=1,2,…$

$$f(x)=\frac{1}{x+a}$$ $$f:\mathbb{R}_{\le0}\rightarrow \mathbb{R}_{\le0},a<0$$ $$f:\mathbb{R}_{=0}\rightarrow \mathbb{R}_{=0},a=0$$ $$f:\mathbb{R}_{\ge0}\rightarrow \mathbb{R}_{\ge0},a>0$$ ...
1
vote
1answer
101 views

Strange result about the log sum

I am working in a infinity sum and I get the strange result $$\sum _{n=1}^{\infty } \frac{1}{2} \log \left(\frac{1}{n^2}\right)=\log (2 \pi )$$ it seem as $$-2 \zeta '(0)$$ but i do not justify? it ...
3
votes
2answers
62 views

How is the Radius of Convergence of a Series determined?

Consider $$\sum_{n=0}^{\infty}\frac{(-1)^nx^n}{(n+1)^2}$$ which by the ratio test the ratio of two consecutive terms converges to $|x|$ as $n\rightarrow \infty$ and has a radius of convergence equal ...
1
vote
3answers
42 views

Evaluating the ratio $ {{a_{n+1}}\over{a_n}}$ in calculating the radius of convergence for a power series

In calculating the radius of convergence for the power series $$ \sum_{n=1}^\infty {{(2n)!}\over(n!)^2}\ x^n $$ By the ratio test, we let $$ a_n = \lvert {{(2n)!}\over(n!)^2}\ x^n \rvert \quad\quad ...
1
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0answers
40 views

Does the “alternating” harmonic series where only prime terms are negative converge?

We know that the harmonic series $\sum \frac{1}{n}$ diverges, yet the alternating harmonic series $\sum \frac{(-1)^n}{n}$ converges. Euler famously gave a proof of the infinitude (and of the ...
2
votes
3answers
65 views

Sum of $ \ \frac{1}{(\ln k)^{\ln k}} \ $

How do I find out if the infinite sum of $ \ \frac{1}{(\ln k)^{\ln k}} \ $ is convergent or divergent? I'm given a hint: $ \ \ln k \ = \ e^{\ln(\ln k)}$ but I can't figure out how to apply that.
0
votes
1answer
11 views

Equidistribution and Smaller Sets

I have a question about equidistribution. In Wikipedia, Equidistributed sequence shows in the equation in the definition that $n$ reaches almost infinity. I was just wanting to make sure if ...
2
votes
1answer
26 views

How many unique numbers can be obtained by adding two numbers from two different sequences?

Let the two integer sequences $\{a_m\}$ and $\{b_m\}$, be defined as: $a_n+D_n=a_{n+1}$ and $b_n=a_n-k$, where $D_n$ may be any natural number (and $D_i$ may or may not be equal to $D_j$), $k$ is an ...
5
votes
3answers
453 views

What is the pre-requisite knowledge for generating my own integer sequence?

I've recently come across the On-Line Encyclopedia of Integer Sequences and I'm completely fascinated by it; something about how easy integers are to grasp and yet how complex the sequences are. I ...
1
vote
3answers
53 views

Geometric sequence problem

Determine the value(s) of k, so that the positive numbers $\log_8(k-1)$, $3\log_8(k-1)$ and $6$ form a geometric sequence (in order given above).
0
votes
3answers
34 views

Arithmetic and geometric sequence

Which two numbers should be placed between -5 and 49 so that the first three numbers form an arithmetic sequence, whereas the last three numbers form a geometric sequence?
0
votes
2answers
67 views

Is the Taylor series of the $f: \mathbb{R} \to \mathbb{R}$ evaluated at $0$ converges pointwise (in the whole $\mathbb{R}$)?

I would like to solve this problem: Consider the $f: \mathbb{R} \to \mathbb{R}$ function which is $n$-times differentiable for any $n \in \mathbb{N}$. Is it true that the Taylor series of ...
1
vote
2answers
77 views

Prove $\lim \frac{3^n + 2^n}{5\cdot3^n + 7\cdot2^n} = \frac{1}{5}$

Prove using limit definition. $$\lim \frac{3^n + 2^n}{5\cdot 3^n + 7\cdot 2^n} = \frac{1}{5} $$ My try: $$\left| {\frac{3^n + 2^n}{5\cdot 3^n + 7\cdot 2^n} - \frac{1}{5}} \right| < \varepsilon ...
1
vote
1answer
20 views

Meaning Behind Mapping from a Compact Subset to Another Set

Suppose I tell you that a set, $A$, is compact and a subset of a metric space. This means that it is closed and bounded and that every sequence in set $A$ has a converging sub-sequence. Then I tell ...
0
votes
2answers
28 views

finding for which $c\in \mathbb{R}$ sequence converges

so i am trying to find for which $c\in\mathbb{R}$ this sequence converges: $a_{1}=c$ and $a_{n+1}=1+\frac{a_{n}^2}{4}$ So i got the basic idea how to do this. First i found the candidate for limit: ...
1
vote
3answers
38 views

Product of first $n$-th prime power integers $+ 1$

I was just playing with prime numbers and then I accidentally found this pattern. Let $p_1\cdot p_2\cdot p_3\cdots p_n$ is the product of first $n$-th prime power integers. Prove that: $p_1\cdot ...
0
votes
4answers
87 views

About the limit $\lim_{n\to +\infty}\frac{n^k}{n!}$ for a fixed $k\in\mathbb{N}$.

Given a natural number $k$ and some real number $\epsilon>0$, I have to prove that there exists a natural number $n$ such that $\frac{n^k}{n!}<\varepsilon$. I tried to develop for ...
2
votes
3answers
319 views

Which is the limit of the sequence $\sum_{i=1}^n \frac{\cos(i^2)}{n^2+i^2}$

I can't find the limit as n $\to$ infinity of the sequence: $$\frac{\cos(1)}{n^2 + 1} + \frac{\cos(4)}{n^2 + 4} + \dots + \frac{\cos(n^2)}{n^2 + n^2}$$ I tried to use the inequality $\cos(n^2) < ...
4
votes
0answers
77 views

Limit of $\frac{1-2+\cdots+(2n-1)-2n}{\sqrt{ (n^2+1)}+ \sqrt{ (n^2-1)}}$

Here is the limit to be calculated : $$\lim_{n\to\infty}\frac{1-2+\cdots+(2n-1)-2n}{\sqrt{ (n^2+1)}+ \sqrt{ (n^2-1)}}$$ The question provides no other information. Doubts What is the domain of ...
1
vote
1answer
33 views

Inverse sum representation of sine

The other day I was playing with functions of the form $$ f(x) = \frac{1}{\frac{1}{a_0(x-b_0)} + \frac{1}{a_1(x-b_1)} + \cdots + \frac{1}{a_n(x-b_n)}} $$ and I found particularly that $$ ...
2
votes
1answer
40 views

Use of restriction to disallow aberrant series

My question concerns restrictions on the exercise of normal algebraic rules. The most well known restriction is the prohibition on division by zero (PDZ). This is justified by various 'proofs' of ...
3
votes
0answers
46 views

Convergence Intervals

Consider the function $$f(x) = \frac{x}{1-x}$$ We know that for $x\in(0,1)$, $$f(x) = x\cdot\frac1{1-x} = \sum_{k=0}^\infty x^{k+1} = \sum_{k=0}^\infty x^{k} - 1$$ Now, notice that: $$\frac{x}{1-x} = ...
-3
votes
2answers
63 views

Show that $\sum_{n=0}^\infty\left(\frac{1}{n+1} - \frac{1}{n+2}\right)= 1$ [closed]

Show that $$ \sum_{n=0}^\infty\left(\frac{1}{n+1} - \frac{1}{n+2}\right)= 1 $$
1
vote
1answer
41 views

Examine the uniform convergence of the series $\sum^{\infty}_{n=1}\frac{1}{\sqrt{x+n}}$ if $x \in [0, \infty]$

Examine the uniform convergence of the series $$\sum^{\infty}_{n=1}\frac{1}{\sqrt{x+n}}$$ if $x \in [0, \infty)$ Which series should I choose in Weierstrass M-test to show that is divergent? ...
3
votes
0answers
42 views

Summing the binomial pmf over $n$, part 2

After the great answers I got to this question, I tried summing a similar-looking series using the same strategies ($k \geq 0, \alpha > 1, p \in (0,1)$): $$ \sum_{n=k}^{\infty} {n \choose k} p^k ...
-2
votes
2answers
73 views

What will be the 100th term of the series: [closed]

If $a_1 = 1$ and $a_{n+1}-3a_n+2=4n$ for every positive integer n, then find the value of $a_{100}$
1
vote
2answers
20 views

Variation of geometric, harmonic and arithmetic means in sequence.

A question I got on my test was - Let ${A}_{1}, {G}_{1}$ and ${H}_{1}$, denote the arithmetic, geometric and harmonic means of two distinct positive numbers. For $n\geq 2$, Let ${A}_{n-1}$ and ...
-3
votes
1answer
79 views

Questions from an olympiad on number theory [closed]

The sum of the infinite series: $$ \frac{1}{2} + \frac{2}{8} + \frac{3}{16} + \frac{5}{32} + \frac{8}{64} + \frac{13}{128} + \frac{21}{256} + \frac{34}{512} +....$$ I am able to find the general term ...
3
votes
0answers
33 views

How can I calaculate this complex series? [closed]

Let $A=\{a+bi|a,b\in \mathbb{Z}\}-\{0\}$ For what $n$ does $\sum_{z \in A} 1/z^n$ $(n\in \mathbb{N})$ converge? If it converges, how can I evaluate it? Thanks in advance.
2
votes
2answers
67 views

For $l^2 (\mathbb{N})$ is $(l^2, d_2)= (l^2, d)$ topologically, where $d$ forms the usual product topology on $\mathbb{R}^{\mathbb{N}}$?

If we define $d(x,y)= \max_{n \in \mathbb{N}}( \min \{2^{-n}, |x_n - y_n| \})$ then does this distance form the same topology on $l^2 ( \mathbb{N})$ (the set of all square-summable real sequences) as ...
0
votes
2answers
42 views

Infinite series of trigonometric ratios

The question is to compute: $$(1+\cos A)+2(1+\cos A)^2 + 3(1+\cos A)^3+\ldots = \sum_{k=1}^{\infty}k(1+\cos A)^k.$$ I tried by setting $1+\cos A=y$, then the serie becomes $$y+2y^2+3y^3+\ldots = ...
0
votes
1answer
39 views

For which values of $p$ does the serie $\sum\limits_{n=2}^\infty\frac{1}{n^p\ln(n)}$ converge?

For which values of $p$ does the serie $\sum\limits_{n=2}^\infty\frac{1}{n^p\ln(p)}$ converge? I'm trying to use the ratio test but I can't get a simple term in which use limit easily enough.
0
votes
0answers
24 views

Find the limit $\lim \int_{(0,1]} f_n \, d\lambda $ where $ \lambda $ is lebesgue measure

Find the limit $\lim \int_{(0,1]} f_n \, d\lambda $ where $ \lambda $ is lebesgue measure and $ f_n(x)=\dfrac{|\cos(x^{-2})|}{x^{1-1/n}} $ for $ x\in (0,1] $ Is there lebesgue integrable function $g$ ...
1
vote
3answers
25 views

$a_n - \frac{a_n - v}{s}$ becomes $v$

Given is following sequence: $a_{n+1} = a_n - \frac{a_n - v}{s}$ I found out that $\forall a_0, v, s \in \mathbb{R}, s>0: \lim\limits_{n \to \infty}a_n=v$ But I do not know why. I tried to ...
2
votes
2answers
52 views

Enumerating the positive rationals without repetition

From Makarov's Selected problems in real analysis Let $r_n$ be defined as follows\begin{cases} r_1=1 & \\ r_{2k}=1+r_k \\ r_{2k+1}=\frac{1}{r_{2k}} & \end{cases} ...
2
votes
1answer
45 views

A proof that $\frac{(2\phi)^n-(-1)^n}{\phi^{2n}-(-1)^n}\cdot\left(2^n-\phi^n\right)\cdot\sqrt5\in\mathbb Q$ for all $n\in\mathbb Z$

During computation of some series (with help of a CAS), at an intermediate step I encountered an expression, that after dropping non-essential parts looks like this:$$\mathcal ...
3
votes
0answers
30 views

Show the convergent of sequences and limit? [duplicate]

a)Show that the following sequence of functions is convergent. $f_{1}(x) = \sin(x)$ $f_{2}(x) = \sin(\sin(x))$ $f_{3}(x) = \sin(\sin(\sin(x)))$ ... $f_{n}(x) = \sin(\sin(\sin(...(\sin(x))))$ ...
2
votes
3answers
70 views

Summing the binomial pmf over $n$

I was trying to work out some bounds for a research problem when I came across the innocuous-looking sum: $$ \sum_{n=k}^{\infty} {n \choose k} p^k (1-p)^{n-k}, \quad k \in \mathbb{N}, \; p \in (0,1)$$ ...
26
votes
1answer
341 views
+50

Conjectured value of a harmonic sum $\sum_{n=1}^\infty\left(H_n-\,2H_{2n}+H_{4n}\right)^2$

There is a known asymptotic expansion of harmonic numbers $H_n$ for $n\to\infty$: $$\begin{align}H_n&=\gamma+\ln n+\sum_{k=1}^\infty\left(-\frac{B_k}{k\cdot n^k}\right)\\ &=\gamma+\ln ...
0
votes
1answer
22 views

How many terms lies between

If a have a series from $n^2$ to $(2n)^2$, i.e. $n^2+(n+1)^{2} +...+ (2n)^2$, how many terms lie between $n^2$ and $(2n)^2$ ? Is it $n+1$ terms, or is it $n$ terms, and how to prove that.
12
votes
3answers
109 views

If $\lim_{n\to\infty} x_{n+1}-\frac12 x_n = 0$ then $x_n\to 0$

Let $x_n$ be a sequence of real numbers such that $\displaystyle\lim_{n\to\infty} x_{n+1}-\frac12 x_n = 0$ Prove that $\lim_{n\to\infty} x_n = 0$ I have a proof if one assumes that $(x_n)$ ...
2
votes
3answers
60 views

Let $A_n=\frac{(n+1)+(n+2)+(n+3)+…+(n+n)}{n}$,$B_n=[(n+1)(n+2)(n+3)…(n+n)]^{1/n}$.If $\lim_{n\to\infty}\frac{A_n}{B_n}=\frac{ae}{b}$

For positive integers $n$,let $A_n=\frac{(n+1)+(n+2)+(n+3)+.....+(n+n)}{n}$,$B_n=[(n+1)(n+2)(n+3)....(n+n)]^{1/n}$.If $\lim_{n\to\infty}\frac{A_n}{B_n}=\frac{ae}{b}$ where $a,b\in \mathbb{N}$ and ...
3
votes
2answers
214 views

Where does this sequence $\sqrt{7}$,$\sqrt{7+ \sqrt{7}}$,$\sqrt{7+\sqrt{7+\sqrt{7}}}$,… converge?

The given sequence is $\sqrt{7}$,$\sqrt{7+ \sqrt{7}}$,$\sqrt{7+\sqrt{7+\sqrt{7}}}$,.....and so on. the sequence is increasing so to converge must be bounded above.Now looks like ...
1
vote
3answers
201 views

How could I find the sum of this infinite series by hand?

$$\sum_{n=1}^{\infty}\frac{(7n+32)3^n}{n(n+2)4^n}$$ Thank you!
4
votes
1answer
65 views

Elementary proof that $\sum_{k=1}^{\infty}{\frac{1}{m^{k^2}}}$ is irrational (for any integer $m > 1$)

I used similar technique as Fourier's proof of irrationality of $e$ https://en.wikipedia.org/wiki/Proof_that_e_is_irrational to show that this series is indeed an irrational number but I was wondering ...