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

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4
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Discrete Laplace transform

Yesterday ago I was reading how the Laplace Transform can be interpreted as the continuous analog of the discrete functional dependance of the power series $$f(x) = \sum a(n) x^n$$ This is to say, $$L\...
6
votes
2answers
4k views

About Banach Spaces And Absolute Convergence Of Series

How to prove the following two assertions: If in a normed space $X$, absolute convergence of any series always implies convergence of that series, then $X$ is a Banach space. In a Banach space, ...
2
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2answers
42 views

Proof that $P(x)=x-\frac{1}3 x^3+\frac{1}{5}x^5-\frac{1}{7}x^7+\cdots$ has radius of convergence $1$

Proof that $P(x)=x-\frac{1}{3}x^3+\frac{1}{5}x^5-\frac{1}{7}x^7+\cdots$ has radius of convergence $1$ First of all, I need to convert this to a series: $$\sum_{k=1}^\infty \frac{x^{2k-1}(-1)^k}{...
2
votes
1answer
15 views

Are all sets of $n$, s.t. $R(m)^n=I$, where $R(m)$ is any sequence of $m$ moves on a Rubik's cube and $I$ is the identity operator, known?

I've written a program that finds the number of times, $n$, one must apply any operation $R_i(m)$, which consists of $m$ single moves/turns/elementary operations on a Rubik's cube, s.t. $R_i(m)^n=I$, ...
4
votes
3answers
47 views

Convergence of a Harmonic Continued Fraction

Does this continued fraction converge? $$\large\frac { 1 }{ 1+\frac { 1 }{ 2+\frac { 1 }{ 3+\frac { 1 }{ 4+\dots } } } } $$ ($[0;1,2,3,4, \dots]$) I tried approximating a few values but I ...
1
vote
3answers
90 views

Show $\sum_{k=1}^{n}\frac{1}{k}\sim \ln(n)$

there is an example of how we apply Integral test for convergence Theorem: Consider an $n_{0}$ and a non-negative, continuous function $f$ defined on the unbounded $[n_0,+\infty[$, on which it ...
5
votes
2answers
113 views

Limits of $a_n, b_n, c_n$

Three given positive $a_1, b_1, c_1$, such that $a_1+b_1+c_1=1, \forall\ n,$ $$a_{n+1}=a_n^2+2b_nc_n, b_{n+1}=b_n^2+2a_nc_n, c_{n+1}=c_n^2+2a_nb_n$$ Prove $\{a_n\},\{b_n\}$ and $ \{c_n\} $ are ...
1
vote
2answers
25 views

Proof of sum convergence?

$$\sum_{n=1}^{\infty} \frac{2\sqrt{n} + 1}{n^2 + n + 1}$$ This seems like a problem that could be handled by the comparison test. So we need an $f(n) > \displaystyle \frac{2\sqrt{n} + 1}{n^2 + n ...
1
vote
3answers
34 views

Prove inequality when $a_{1},a_{2},…,a_{n}$ are positive numbers.

Let's assume that $a_{1},a_{2},...,a_{n}$ are positive. How to prove this inequality: $(a_{1}+a_{2}+...+a_{n})(\frac{1}{a_{1}}+\frac{1}{a_{2}}+...+\frac{1}{a_{n}})\geq n^{2}$ My effort: I don't know ...
-1
votes
0answers
22 views

On a certain limit

Let $t \in R$ and $0<\alpha<\beta <1$. For $m \in N$, $S_m$ is defined as follows: $$ S_m= \sum_{1 \le k < l \le m}(-1)^{k+l} \cos(t\ln(k/l)) (\frac{1}{k^{\alpha}} - \frac{1}{k^{\beta}}...
3
votes
1answer
49 views

Product of Power Series of Different Powers

I am trying to find the product $M$ of two power series of the form \begin{equation} M=\left(\sum_{n=0}^{\infty}a_{n}\, x^{2n}\right) \left(\sum_{n=0}^{\infty}b_{n}\, x^{n}\right) \end{equation} ...
1
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0answers
45 views

Convergence of a sequence (hard)

Bonjour, with the hypothesis that: $\sum \limits_{k=1}^{+\infty} (-1)^{k+1} k^{-a} \sin (b\ln k) = 0$ where $0 < a < 1$ et $b > 0 $. Can we say that, for $ -1 < \epsilon < 1$, $\...
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0answers
26 views

How to prove this sequence of functions is Cauchy for this metric?

Let $C([-1,1])$ be the set of all continuous functions from $[-1,1]$ to $\mathbb{R}$. Let $d_1$ be a metric on this space defined as $$d_1(f,g) = \int_{-1}^{1} |f(t) - g(t)| dt. $$ Now consider for ...
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votes
2answers
286 views
+50

The relationship between tan(x) and square roots

Please note: I am working in DEGREES I think the easiest way to illustrate my point is by showing some examples: $ \tan(0^\circ) = \sqrt 0 = 0$ $ \tan(22.5^\circ) = \sqrt 2 -1$ $ 3 \cdot \tan(30 ^...
4
votes
2answers
90 views

Writing continued fractions of irrational numbers as infinite series

Infinite sums have been formulated for famous irrational numbers, such as $\pi, \phi,e,\sqrt2$ and a few others that can be listed here and here: Here are some examples: (There are more examples ...
1
vote
1answer
53 views

Limit of sequence with floor function of sqrt problem

I have the following sequence and limit: $$\left(a_n\right)=\left(n-\sqrt{n}\left\lfloor{\sqrt{n}}\right\rfloor\right)$$ $$\lim_{n\to\infty}{n\left(\sqrt{n^2+2\ell}-n\right)}=\ell,$$ where $\ell$ is ...
0
votes
2answers
59 views

Finding a closed form solution

For the sequence 0 2 8 34 144 ... The recurrence relation is: $$\begin{align*} E(n) = 4*E(n-1)+E(n-2) \end{align*}$$ How to calculate the closed form expression ...
14
votes
1answer
177 views

Evaluating $\sum_{n=1}^\infty\frac{1}{n \binom{kn}{n}}$

For $k=1$, the series does not converge. When $k=2$, I can prove that: $$\sum_{n=1}^\infty\frac{1}{n \binom{2n}{n}}=\frac{\pi}{3\sqrt{3}}$$ Using the result of: $$\int_0^\infty \frac{x^ndx}{(x+1)^{...
1
vote
1answer
24 views

Is the mathematical syntax correct here (Taylor-polynomial)?

Say I'm supposed to create the $2^{nd}$ degree Taylor-polynomial of $f(x) = \cos x$ at $x_{0} = 0$ I'd like to know if the syntax is correct, how I solved this little task. We have defined the ...
0
votes
1answer
35 views

Is $g$ an injective continuous mapping?

For an artibtrary $t \in \mathbb R $, let's define a mapping $$g: (0,1) \to R^2$$ as follows: $$g(\alpha)= \left( \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^{\alpha}}\cos(t\ln(n)), \sum_{n=1}^{\...
-1
votes
0answers
79 views

What is the logic to the sequence $10$, $2$, $4$, $6$, $2$, $8$, $5$? [on hold]

$$10, 2, 4, 6, 2, 8, 5$$ Can someone tell me the logic behind this sequence? It was a question on a magazine without the last number —$5$— but I can't find any logic.
0
votes
2answers
48 views

Absolutely Convergent, Conditionally Convergent or Divergent?

$$\sum_{k=0}^{\infty} (-1)^{k+1}\frac{\sqrt{k}}{k+1}$$ This problem is asking me to prove if this series is absolutely convergent, conditionally convergent or divergent, but I don't know how to start ...
0
votes
3answers
66 views

Infinite sum of sequence

I was just curious to know how to find the infinite sum of the below sequence. $$\frac{1}{4}+\frac{2}{4^2}+\frac{1}{4^3}+\frac{2}{4^4}+\frac{1}{4^5}+\frac{2}{4^6}+\frac{1}{4^7}+\dots$$ I know it can ...
0
votes
0answers
22 views

Choosing values in a strong induction

The sequence s0,s1,s2... is defined by s0=1 and for all integers n>0, $s(n)=s(⌊n/2⌋)+s(⌊2n/5⌋) + n.$ Prove, using strong induction, that S(n) > 4n for all integers n>=3. To my knowledge, I only have ...
0
votes
2answers
40 views

Nth term Fibonacci formula.

I'm studying power series and in the video the professor showing an example of finding formula for n term of Fibonacci sequence. at the middle of the video (6:00) https://www.youtube.com/watch?v=CR-...
6
votes
3answers
229 views

Seeking closed form for infinite sum $\sum \limits_{ n=1 }^{ \infty }{ \frac { { \left(n! \right) }^{ 2 } }{ { n }^{ 3 }(2n)! } }$

$\displaystyle \sum _{ n=1 }^{ \infty }{ \frac { { \left(n! \right) }^{ 2 } }{ { n }^{ 3 }(2n)! } }$ is approximately $.5229461921333351$ but I've been assured that there is a closed form for this ...
0
votes
1answer
28 views

Difference between little o and big O notation in taylor expansion

I know I can say that: $$ y(t+\Delta t)=y(t)+y'(t)\Delta t+o(\Delta t) $$ But can I say that: $$ y(t+\Delta t)=y(t)+y'(t)\Delta t+O((\Delta t)^2) $$ In both cases, do I have to add that $\Delta t \...
2
votes
0answers
50 views

Show that: $\lim_{n \to \infty}\prod_{k=1}^n \cos(k\sqrt{\frac{3}{n^3}} t) = e^{- \frac{t^2}{2}}$ [duplicate]

Is it true that $$\lim_{n \to \infty}\prod_{k=1}^n \cos\left(k\sqrt{\frac{3}{n^3}} t\right) = e^{- \frac{t^2}{2}}$$ ? How to proceed ?
4
votes
1answer
57 views

Find the value of : $\lim_{n\to\infty} \left( \left(\sum_{k=n+1}^{2n}2\sqrt[2k]{2k}-\sqrt[k]{k}\right)-n\right)$

Find $$\lim_{n\to\infty} \left( \left(\sum_{k=n+1}^{2n}2\sqrt[2k]{2k}-\sqrt[k]{k}\right)-n\right).$$ I have tried rewriting the sum in a clever way, applying the Mean Value Theorem or Stolz-Cesaro ...
17
votes
3answers
1k views

Find the value of : $\lim\limits_{n\rightarrow\infty}\left({2\sqrt n}-\sum\limits_{k=1}^n\frac{1}{\sqrt k}\right)$

How to find $\displaystyle\lim_{n\rightarrow\infty}\left({2\sqrt n}-\sum_{k=1}^n\frac{1}{\sqrt k}\right)$ ? And generally does the limit of the integral of f(x) minus the sum of f(x) exist? How to ...
4
votes
1answer
78 views

Find the value of : $\lim_{n \rightarrow \infty} \prod_{k=1}^n \left(1+\ln\left(\frac{k+\sqrt{k^2+n^2}}{n}\right)^{\frac{1}{n}}\right)$

Compute $\displaystyle \lim_{n \rightarrow \infty} \prod_{k=1}^n \left(1+\ln\left(\frac{k+\sqrt{k^2+n^2}}{n}\right)^{\frac{1}{n}}\right)$ Note that $\frac{k+\sqrt{k^2+n^2}}{n}\geq 1$ so we're ...
3
votes
0answers
111 views

conjectured arithmetic properties of some continued fraction

Given the continued fraction found in this post,bearing a striking resemblance to the one in this post $$G(q)=\cfrac{1}{1-q+\cfrac{q(1-q)^2}{1-q^3+\cfrac{q(1-q^2)^2}{1-q^5+\cfrac{q(1-q^3)^2}{1-q^7+\...
0
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2answers
23 views

Trouble in constructing a sequence.

How can I construct a sequence $\{x_n\}$ of rational numbers such that it converges to a point $a$ which is irrational? Please help me.Thank you in advance.
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3answers
38 views

Problem in finding an example related to infinite series.

The problem is : Give an example of a divergent series $\sum u_n$ such that $\sum u_{3n}$ is convergent. Please help me in finding this example.Thank you in advance.
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0answers
19 views

Maximum of derivative hierarchy

Given a function $a(x)$, is $\displaystyle\max\left(\frac{1}{2}\frac{da(x)}{dx}\right) > \max\left(\frac{1}{3!}\frac{d^2a(x)}{dx^2}\right) > \max\left( \frac{1}{4!}\frac{d^3a(x)}{dx^3}\right) &...
0
votes
1answer
86 views
+100

Having no derangements — any advantage?

Is there any problem-solving advantage when a sequence has no derangements? Edit Perhaps the non-derangements are just a by-product of the argument that we find primes in $(m-k,k] \iff k|m$? In an ...
1
vote
1answer
24 views

Uniformly Cauchy sequence of functions

I am trying to show the following: For each $n \in \mathbb N$, let $f_n:X \to Y$, where $(Y,d)$ is a complete metric. Suppose that for every $\epsilon>0$, there exists $n_0 \in \mathbb N$ such ...
3
votes
2answers
89 views

Prove that $\lim a_{n}=e$?

Given that $a_{1}=0$, $a_{2}=1$ and $$a_{n+2}=\frac{(n+2)a_{n+1}-a_{n}}{n+1}$$ prove that $\lim\limits_{n\to\infty} a_n=e$ What I did: It was hinted to prove that $a_{n+1}-a_{n}=\frac{1}{n!}$ ...
1
vote
3answers
110 views

Coefficients in series expansion of $\left(\frac{x}{1-x}\right)^3$

How can I find the coefficient of $x^m$ in $$ \left(\sum_{n=1}^{\infty}x^n\right)^3 $$ for some $m\in{\Bbb N}$? I attempted it using the Cauchy products, but things got messy. I think one might want ...
3
votes
3answers
59 views

Taylor-polynomial of function $f(x) = e^{x}*\sin(2x)$

This is not homework, I'm asking to learn for an exam which I'll write in 2.5 months. Count the Taylor-polynomial 3th grade of the function $f: \mathbb{R} \rightarrow \mathbb{R}, f(x) = e^{x}*\sin(...
0
votes
1answer
50 views

When does a sequence (or a series) of real-analytic functions converge to a real-analytic function?

It is well known that a sequence (or a series) of holomorphic functions converging uniformly converges to a holomorphic function. I would like to know under what condition a sequence (or a series) or ...
2
votes
2answers
31 views

Series with terms indexed by transfinite ordinals

I learned this past year how to deal with summation of infinite series $a_0 + a_1 + a_2 + ...$ with indices running over the natural numbers. I've also learned about what comes "after" the natural ...
0
votes
1answer
54 views

complex sequences

my series and sequence knowledge has gone a little rusty so I was wondering if you could help me on the right path here. The assignment is to calculate the sum of the series $(\frac{1}{8})^n e^{j(n{\...
0
votes
1answer
21 views

How to determine when a rounded sequence “converges” and what the convergence value is

I have a process on a server that iteratively computes a value over time. The value follows a fairly simple formula, which would generate a convergent sequence; however, the value of the formula is ...
33
votes
3answers
3k views

Why is this series of square root of twos equal $\pi$?

Wikipedia claims this but only cites an offline proof: $$\lim_{n\to\infty} 2^n \sqrt{2-\sqrt{2+\cdots+ \sqrt 2}} = \pi$$ for $n$ square roots and one minus sign. The formula is not the "usual" one, ...
1
vote
0answers
56 views

nature of series $\sum_{n\geq 0}(-1)^{n}u_n $

Let $(u_n)_{n\in\mathbb{N}}$ be sequence defined as follows: $$\left\{ \begin{cases} u_0\in\mathbb{R}^{+}\\\forall n\in\mathbb{N},\quad u_{n+1}=\dfrac{e^{-u_n}}{n+1}\\ \end{cases} \right\}$$ ...
1
vote
2answers
17 views

Evenly filling spaces for a specific average value

Imagine I have $N$ spaces. Each space can be empty, or occupied. Given a fixed point value $x$ between zero and one, I would like to evenly populate the $N$ spaces such that $\frac{N_{\text{occupied}...
6
votes
2answers
62 views

Deriving sum of powers formula using generating functions

Just for fun I wanted to try to derive a formula for the sum of $p$-powers using generating functions, but without using any literature or websites for help. However I do need a small push or hint. ...
11
votes
2answers
190 views

Sum of $\sum \limits_{n=0}^{\infty} \frac{1}{(kn)!}$

Does a closed form exist for $$\sum \limits_{n=0}^{\infty} \frac{1}{(kn)!}$$ in terms of $k$ and other functions? The best that I have been able to do is solve the case where $k=1$, since the ...
1
vote
1answer
41 views

Partial sum of bounded series is Cauchy in $C(X, \mathbb{R})$?

I am reading a proof of Tietze's Extension Theorem and there was a claim that, given a sequence of functions $h_n(x) : X \to \mathbb{R}$ If $$G = \sum\limits_{n = 1}^\infty h_n(x)$$ Is bounded, then ...