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

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1answer
57 views

Is the calculation of the series in this video correct?

I am watching this video (from MIT OCW) and Prof. Jerison is explaining about series. He is trying to calculate that if some blocks of equal length are kept on top of each other, will the last block ...
1
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2answers
39 views

Taylor series convergence for sin x

a. $\forall x\in(0,\pi/2),\quad x-\frac{x^3}{3!}<\sin x<x-\frac{x^3}{3!}+\frac{x^5}{5!},$ b. $\forall x\in(0,\pi/2),\quad x-\frac{x^3}{3!}+\frac{x^5}{5!}+\cdots-\frac{x^{4k-1}}{(4k-1)!}<\sin ...
1
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1answer
32 views

For what p is the series abolsutely convergent and conditionally convergent?

My lecturer has a passion for logs, and I'm reviewing some of her past papers and I found this question and I'm having a quite a difficult time dealing with, any help would be appreciated $$ ...
0
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2answers
52 views

convergence of sin functions absolutely for all x in $\mathbb R$ [closed]

How do I show that $\sum_{n = 1}^{\infty}2^n \sin(x/3^n)$ converges absolutely for all $x \in \mathbb{R}\;$?
9
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5answers
242 views

Prove that $\int_0^1\frac{1-x}{1-x^6}\ln^4x\,dx=\frac{16\sqrt{3}}{729}\pi^5+\frac{605}{54}\zeta(5)$

This integral comes from a well-known site (I am sorry, the site is classified due to regarding the OP.) $$\int_0^1\frac{1-x}{1-x^6}\ln^4x\,dx$$ I can calculate the integral using the help of ...
1
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1answer
85 views

Explanation of the formulas for sums $\sum nr^n$ and $\sum n^2 r^n$

So I am studying series for an exam right now and there is an example in the book I am studying (unfortunately the book is specific to my university so I cannot give any link) where certain series' ...
3
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2answers
65 views

Radius of convergence of power series which has factorial term

I am trying to find radius of convergence of the following power series: $\sum_{n\geq 1} n^n z^{n!}$ I tried ratio test but it became complicated, I have never seen such radius of convergence problem ...
1
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0answers
69 views

Why is sequence $(1+\frac{1}{n})^{n+1}$ descending? [duplicate]

I was studying the proof of $e$ number when I noticed something: Why is the sequence $(1+\frac{1}{n})^{n+1}$ descending? It starts ascending with grater n but in one moment it starts descending? Why ...
1
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1answer
44 views

Series converges but term by term integration not valid?

Give an example of a series $\sum g_n$ of Lebesgue integrable functions on $\mathbb{R}$ that converges but for which term by term integration is not valid. This is last minute exam revision so I do ...
3
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0answers
51 views

Analytic solution for system of Digamma function equations

Peace be upon you, There is a while, I am seeking for the analytic solution of $\alpha$ and $\beta$ in this system \begin{align*} \begin{cases} \psi(\alpha)-\psi(\alpha+\beta)=c_1\\ ...
0
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1answer
29 views

Writing dense sets in terms of set of integers

Can we write every dense set in $\mathbb R$ as {$x_n$}$\mathbb Z$ , where {$x_n$} is a real sequence with limit $0$ ?
2
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3answers
100 views

Show that $q^4+2pq^2 +p^2 = 2pq -(pq)^2 -1$ becomes $p^3+q^3+3pq-1=0$.

I know that these two are exactly the same equation but I can't seem to prove it. You are also given that $p+q=1$. This is a follow up from a similar question.
0
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1answer
58 views

Rate of convergence of an iterative root finding method similar to Newton-Raphson

We are defining an algorithm as follows: Let $f(x)$ be a function with a root in $[a,b]$. We define a series $\{x_k\}_{k=1}^{\infty}$ as follows: $x_{k+1}=x_k-f(x_k)\frac{b-a}{f(b)-f(a)}$. ...
2
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2answers
156 views

Prime Number Sum Sequence (Amateur)

SOLVED: This is false Beginning with 3, add the next consecutive prime (2) and then take that sum (5) and add that to next consecutive prime (3) to get (8), and so on... $$ 3 + 2 = 5 $$ $$ 5 + 3 = 8 ...
1
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0answers
62 views

sum of reciprocal power-1

I found this in my old notebook $$\sum_{n \text{ perfect power}} {\frac{1}{n-1}} = 1$$ and this was my "proof" $$ \begin{align} \frac{1}{1}+\frac{1}{2}+\cdots ...
1
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2answers
31 views

Divergence of a recursive sequence

If $(x_n)$ isthe sequence defined by $x_1=\frac{1}{2}$ and $x_{n+1}=\sqrt{x_n^2 +x_n +1}$, show that $\lim x_n = \infty$ Ive tried a couple of things but none of them helped. Ive tried to suppose, by ...
0
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1answer
25 views

Count of paths between N points

I am trying to arrive at a formula that will give me the number of distinct paths between a set of discrete points on a map. I have worked out that I can calculate it using a series of additions: ...
24
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1answer
534 views

Prove that $\displaystyle\int_0^1{\left\lfloor{1\over x}\right\rfloor}^{-1}\!\!dx={1\over2^2}+{1\over3^2}+{1\over4^2}+\cdots.$

Question. Let $f:[0,1]\to\mathbb R$ given by $$ f(x)=\left\{\,\,\, \begin{array}{ccc} \displaystyle{\left\lfloor{1\over x}\right\rfloor}^{-1}_{\hphantom{|_|}}&\text{if} & 0\lt x\le 1, \\ ...
0
votes
1answer
27 views

Sequence of sets $f_n$ that converge almost everywhere to $f$ but not almost uniformly to $f$?

I've been trying to think of a simple example of a sequence of sets $f_n$ that converge almost everywhere to $f$ but not almost uniformly to $f$. Suppose $f$ is the zero function for the sake of ...
0
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2answers
37 views

How to use the Comparison Test to investigate the convergence of $\sum (\ln n)/n^\alpha$?

Let $$\sum\limits_{n=1}^\infty \frac{\ln n}{n^\alpha}, \alpha\in\Bbb{R}$$ I need to investigate the convergence of this series. I've read that since the series is positive for all $n$ then it ...
0
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5answers
57 views

Arithmetic Progression.

Q. The ratio between the sum of $n$ terms of two A.P's is $3n+8:7n+15$. Find the ratio between their $12$th term. My method: Given: $\frac{S_n}{s_n}=\frac{3n+8}{7n+15}$ ...
0
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1answer
45 views

Show that C is a closed convex subset and its element of minimum norm

I have a lot of problems with the following exercise that I can't solve. Let $(L^1((0,1)), \|\cdot\|_{L^1}):=(E,\|\cdot\|)$ and $$C:=\{u\in E:u\geq0 \text{ a.e } x\in (0,1),\quad T(u)\geq 1\},$$ ...
1
vote
1answer
58 views

What is the closed form for this sequence, powers of $4$?

What is the closed form for this sequence: 1, 4, 12, 40, 148, 576, 2284, 9112, 36420, 145648, 582556, 2330184, 9320692, 37282720, 149130828, 596523256, 2386092964, 9544371792, ...
9
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1answer
165 views

Study of the convergence of a sequence with repeated radicals

Consider the sequence $$ a_n = \sqrt {1!\sqrt {2!\cdots\sqrt {n!} } }, \quad n\in\mathbb N. $$ Does this sequence converge? Clearly, $a_n$ is monotonically increasing. Therefore, there are two ...
4
votes
2answers
459 views

Determine the convergence of $\sum_{n=1}^{\infty}\left(1-\cos\frac{1}{n}\right)$

I'm having trouble determining the convergence of the series: $$ \sum_{n=1}^{\infty}\left[1-\cos\left(1 \over n\right)\right]. $$ I have tried the root test: ...
0
votes
3answers
59 views

How to show that $a_{n+1} = \frac{a_n^2 +3}{4} $ is increasing.

I'm not good at finding whether a sequence is increasing or decreasing. $a_{n+1} = \dfrac{a_n^2 +3}{4}$ is the recursive sequence where $ a_1 =0$ How to get the approach to do something like this? ...
4
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1answer
64 views

By plugging $p=1-q$, into the $3$ equations show that $x=y=z$

By plugging $p=1-q$, into the 3 equations: $$\begin{cases} z=py+qx \\ x=pz+qy \\ y=px+qz \end{cases}$$ show that $\boxed{x=y=z}$ This is from the final part of question 7 in this STEP paper, and is ...
0
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0answers
25 views

A sequence $a_n$ converges to $+\infty$ iff it has $+\infty$ as it's only partial limit

In one of my books there was an exercise to prove that: A sequence $a_n$ converges to $+\infty$ iff it has $+\infty$ as it's only partial limit (I know a sequence doesn't really converge to ...
6
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1answer
353 views

Approximating the exponential function

I have found experimentally something that seems graphically like an approximation of the exponential function. However, it is totally experimental and I have no idea whether it really converges ...
5
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2answers
194 views

Dynamics of the repetitions of $f(z) = z^{2} +\frac{1}{4}$

This is probably a classic and most likely an easy question, but I was not able to find an answer. If we have the iteration $z_{n+1}=g(z_n)$, where $$g(z) = z + a z^{2},$$ with $a\ne 0$, then using a ...
2
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3answers
95 views

How to prove that $\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}$ without using Fourier Series [duplicate]

Can we prove that $\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}$ without using Fourier series?
4
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2answers
63 views

Prove $A=\{x\in (1,2): \text{the decimal expansion of $x$ contains only } 1,3 \text{ or }5\}$ is compact

Prove $A=\{x\in (1,2): \text{the decimal expansion of $x$ contains only}~ 1,~3~\text{or}~5\}$ is compact It is bounded being a subset of $(1,2)$ the only thing left to prove is that it is closed. ...
16
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1answer
493 views

Simplify $\left({\sum_{k=1}^{2499}\sqrt{10+{\sqrt{50+\sqrt{k}}}}}\right)\left({\sum_{k=1}^{2499}\sqrt{10-{\sqrt{50+\sqrt{k}}}}}\right)^{-1}$

Simplify $$\frac{\sum_{k=1}^{2499}\sqrt{10+{\sqrt{50+\sqrt{k}}}}}{\sum_{k=1}^{2499}\sqrt{10-{\sqrt{50+\sqrt{k}}}}}$$ I don't have any good idea. I need your help.
3
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1answer
53 views

Do I need to use induction to have sufficient rigor in this proof?

I'm taking my first analysis class this summer. The professor asked us to prove that $a^{2n}-b^{2n}$ is divisible by $a+b$. After dorking around with the first couple of $n$ I was able to come up ...
33
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1answer
690 views

Why does this ratio of sums of square roots equal $1+\sqrt2+\sqrt{4+2\sqrt2}=\cot\frac\pi{16}$ for any natural number $n$?

Why is the following function $f(n)$ constant for any natural number $n$? $$f(n)=\frac{\sum_{k=1}^{n^2+2n}\sqrt{\sqrt{2n+2}+{\sqrt{n+1+\sqrt ...
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4answers
48 views

Show that a sequence is bounded if and only if there exists a K $\in\mathbb{R}$ such that $|\ a_n\ | \leq K$ $\forall n\in \mathbb{N}$.

Show that a sequence is bounded if and only if there exists a K $\in\mathbb{R}$ such that $|\ a_n\ | \leq K$ $\forall n\in \mathbb{N}$. Is it correct/enough for me to show the following: Let ...
0
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2answers
23 views

Closing the gap between convergence and divergence of $\sum_n 1/n^{1 + 1/f(n)}$ for increasing $f$.

So I have managed to show that $\sum_n 1/n^{1 + 1 / \log n}$ diverges and $\sum_n 1/n^{1 + 1 / \log \log n}$ converges. But the growth rates of $\log n$ and $\log \log n$ are very different. Can ...
0
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5answers
41 views

Does $\sum_{i=1}^\infty \frac{1}{(a+i)^b}$ converge for $b>1$?

Does $$\sum_{i=1}^\infty \frac{1}{(a+i)^b}$$ converge for $b>1$? What is the name of this series?
0
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0answers
24 views

Convergence of Newton series

What is the condition for a real valued function of a real variable to have a Newton series which converges to that function pointwise? It feels like there should be a condition similar to that for ...
14
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1answer
551 views

Do runs of every length occur in this string?

In reference to the strings defined here (constructed by repeatedly appending the last "half" of the current string), consider the particular infinite string $s$ generated by starting with ...
3
votes
1answer
50 views

Limit of a sequence of averaged numbers?

Let $a_0 = 0$, $a_1 = 1$, and $a_n = \frac{a_{n-1}+a_{n-2}}{2}$ for all $n \ge 2$. Consider $\lim \limits_{n \to \infty} a_n$. Using a quick python script I found that for large $n$ $a_n$ tends to ...
1
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1answer
91 views

Limit of quotient of two similar series

Let $0 < p < 1,0 < q < 1$. Find the limit $L = \mathop {\lim }\limits_{n \to \infty } \left( {1 - p} \right)\frac{{\sum\limits_{m = 0}^\infty {{{\left( {1 - p} \right)}^m}{{\left( {1 - ...
0
votes
1answer
22 views

Recurrence relation with geometric sequence

Reading the seemingly excellent book Basic Stochastic Processes (Brzezniak), but got confused by a derivation, page 86-87 specifically. We arrive at this formula: $x_{n+1} - \frac{q}{p+q} = ...
0
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1answer
58 views

Series problem:

Parts i) & ii) I can solve. For part iii) I get $z=py+qx$ [For $n=0$] $x=pz+qy$ [For $n=1$] $y=px+qz$ [For $n=2$] leading to $(1-pq)x=(q^2+p)z$ [1] $(1-pq)z=(q^2+p)y$ [2] ...
0
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2answers
65 views

Sequential Algebraic Problem:

The first part i) I can do. For part ii) this is how far I can get: If n is odd then $y=px+qy$ If n is even then $x=py+qx$ After some rearranging i end up with $y(1-q)=px$ & $x(1-q)=py$ and ...
5
votes
5answers
257 views

Prove that the sequence with $T(0)=1$ and $T(n) = 1 + \sum_{j=0}^{n-1}T(j)$ is given by $T(n)=2^n$

$T(0)=1 \\ T(n) = 1 + \sum_{j=0}^{n-1}T(j) \\ $ Show that $T(n) = 2^n$. I know how to prove this by induction, but I would like to know how to show this using first principles. Edit: The way I want ...
2
votes
1answer
63 views

If $ \sum a_n$ diverges and $\lambda_n \to \infty$, does the series $ \sum \lambda_na_n$ diverge?

Suppose that the series $\displaystyle \sum a_n$ diverges and $\lambda_n \to \infty$. Does the series $\displaystyle \sum \lambda_na_n$ diverge? And what happens if $\{\lambda_n\}$ is an unbounded ...
11
votes
1answer
258 views

Another Ramanujan's formula dealing with $\coth^{2}(5\pi)$

Ramanujan mentions in one of his letters to Hardy that $$\frac{1^{5}}{e^{2\pi} - 1}\cdot\frac{1}{2500 + 1^{4}} + \frac{2^{5}}{e^{4\pi} - 1}\cdot\frac{1}{2500 + 2^{4}} + \cdots = ...
1
vote
7answers
111 views

Error in proving of the formula the sum of squares

Given formula $$ \sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6} $$ And I tried to prove it in that way: $$ \sum_{k=1}^n (k^2)'=2\sum_{k=1}^n k=2(\frac{n(n+1)}{2})=n^2+n $$ $$ \int (n^2+n)\ \text d ...
0
votes
2answers
35 views

What does this sequence converge to?

Could you please help me solve this question relating to sequences? Suppose that a sequence $\{a_n\}$ converges to $\pi$. Then the sequence $\{\cos(a_n)\}$ _____. The answer is converges to $-1$ ...