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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3answers
45 views

The second term of an arithmetic sequence is $13$ and $5^{th}$ term is $31$. What is the $17^{th}$ term of the sequence?

$4.$ Evaluate the following: $a.$ The second term of an arithmetic sequence is $13$ and $5^{th}$ term is $31$. What is the $17^{th}$ term of the sequence? I am trying to do this question (see ...
1
vote
1answer
16 views

Fourier coefficients of $\sin x$ from $-π/2\le x\le π/2$

I need to find the fourier coefficients of $\sin x$ from $-π/2\le x\le π/2$, but I have a doubt, is the period $T=\pi$ or $T=2\pi$? And the other doubt is: this is an odd function, right? So, $a_0=0$, ...
-4
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1answer
33 views

Convergence of a series, does it converges? [closed]

Determine if this series converges, and if converges, compute its sum: $$ \sum_{n=1}^{\infty}2^{3n}4^{2-2n}. $$ Determine if the series converges or diverges: $$ \sum_{n=1}^{\infty}\cos\frac{n^{2}+1}...
1
vote
2answers
63 views

Convergence of Hyper-Geometric Function at $x=1$

I know that the Hyper-Geometric function is given by $$_2F_1(a,b,c,x)=\sum_{i=0}^{\infty}h_nx^n=\sum_{i=0}^{\infty}\frac{(a)_n(b)_n}{n!(c)_n}x^n \tag{1}$$ I want to know that under what conditions ...
3
votes
1answer
87 views

Computed wrong the sum $\displaystyle\sum_{n\ge1}\frac{\cos n}{n^2} $ [duplicate]

I've computed (using standard complex analysis techniques) the sum $$ \sum_{n\ge1}\frac{\cos n}{n^2} $$ and I found $\pi^2/6+1/4$ which is strictly greater than $\pi^2/6$, and this is impossibile, ...
0
votes
1answer
21 views

Convergence of random variable 5

If $Q < \frac{n}{m^2} X_n$ where $X_n$ is a sequence of random variables, $X_n \xrightarrow{a.s}1$, $0\leq Q \leq1$, $m=\omega(\sqrt{n})$ (The $\omega$ denotes the order, see here). Then, how can ...
0
votes
1answer
25 views

Number of Arithmetic Means inserted between 2 quantities

If $n$ Arithmetic Means are inserted between two quantities $a$ and $b$, then their sum is equal to what ? What's the formula for these type of questions ? And can you please explain how it's derived ...
2
votes
1answer
28 views

Complex Fourier series and half-range expansions

I need to find the complex Fourier series for $f(x) = x$, where $0 < x < 2\pi$. I tried to solve this in two different ways, first with even extension, and then with odd, but I did not get the ...
-1
votes
2answers
55 views

Sum of $\infty$ terms in Geometric Progression

Please help me to do this question If $3+3a+3a^{2}+.....\infty = \frac {45}8$ , then what is the value of $a$ ? My Approach to the Problem :- $3(1+a+a^{2}+.....\infty) = \frac {45}8$ $1+a+a^{2}+......
1
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1answer
39 views

Prove the convergence of sequence and find its limit

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ with: $$f(x) = \frac{1}{x^2}e^{\frac{1}{x}}$$ We consider the sequence $x_n$, having $x_0 \in \left ( 0, \frac{1}{2} \right )$ and $x_{n+1} = f(\frac{1}{...
2
votes
6answers
293 views

Prove that the derivative of $x^w$ is $w x^{w-1}$ for real $w$

Can anyone give a rigorous proof of the derivative of this type of function? Specifically showing, $\frac{d(x^w)}{dx} = wx^{w-1}$ for a real $w$? I tried to use the Taylor series expansion for $(x+...
2
votes
1answer
94 views

Can you help me to evaluate $\int_{-\infty}^\infty\frac{x^2}{-1+\cosh (2x)}dx$ as $\pi^2/6$? And do you find a similar integral for $\zeta(4)$?

I was inspired in the shape of the integrals for $\zeta(2)$ in A. Córdoba, Encounters at the interface between Number Theory and Harmonic Analysis, Proceedings of the Segundas Jornadas de Teoría de ...
1
vote
1answer
48 views

Find the next number here

27 , 49 , 45 , 100 , 65 , ? The options given are: 329, 225, 324, 400 All I can conclude is that the answer is a square of some number, but I can't seem to find it.
0
votes
1answer
38 views

Does the Taylor series of $e^{f(x)}$ converge everywhere?

In STAT 110, the professor says "the Taylor series of $e^x$ converges everywhere, and then proceeds to convert: $${e}^{t^2/2} = \sum_{i=0}^\infty \frac{{(t^2/2)}^n}{n!}$$ I understand that the ...
1
vote
2answers
69 views

Closed-form Solution for series involving incomplete Gamma Function

I am working on a solution for an intgeral that leads to a series that I am stuck at. Below is what I have done and how I got to the final series. Any ideas on how to solve the series at the end? \...
1
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1answer
62 views

Computing the Series the looks cute

How do we calculate without residues that , $ \displaystyle \sum_{-\infty}^{\infty} \frac{(-1)^n}{(n+a)^6}$ where $ a \in \mathbb{R} $ and $a$ is not an integer ?
0
votes
0answers
60 views

The sequence of such a squares

I don't find this sequence at oeis: a(n) is the smallest square that is the concatenation of exactly n distinct non zero squares (no more and less). I found the first $3$ terms of the sequence : $1,49,...
4
votes
5answers
388 views

Negation of the definition of limit

A sequence $(x_n) $ of real numbers converges to a real number $ x $ if For all $\epsilon> 0 $ there exists a natural number $ n_0 $ such that for all $ n \ge n _0 $, $|x_n - x| < \epsilon $. ...
0
votes
1answer
35 views

Two sequences have the same limit

Let $f$ and $g$ be real-valued continuous functions on $\Bbb R^2$ that satisfy the following condition: $$ x<y \implies x< f(x,y) < g(x,y) <y $$ Assume that there are two sequences $\{a_n\...
5
votes
5answers
1k views

Approximation of log(n!)

I just finished calculus 1 (derivative and integral) then I take another course on calculus 2. In the video the professor talks about the the series $$\frac{n!}{(\frac{n}{e})^n}$$ He shows the ...
1
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1answer
70 views

Is it possible to identify this sequence?

Interested by this question, $j$ being a positive integer, I tried to work the asymptotics of $$S^{(j)}_n=\sum^{n}_{k=0}\frac{\binom{n}{k}}{n^k(k+j)}=\frac{\, _2F_1\left(j,-n;j+1;-\frac{1}{n}\...
0
votes
2answers
60 views

Help determining if the series $\sum_{n=0}^{\infty }\frac{n^2}{3^n}$ converges or diverges [duplicate]

$\sum_{n=0}^{\infty }\frac{n^2}{3^n}$ I've tried using the ratio test but I can't seem to figure out what I'm doing wrong: $$\lim_{n \rightarrow \infty}\left|\frac{(n+1)^2}{3^{n+1}}\cdot \frac{3^n}{...
2
votes
1answer
129 views

A couple of series questions that I just can't figure out (Calc 2)

Show that $$ \begin{align} \left(\frac{\pi}{2}\right)^2\left[\int_0^{\pi/2}\cos^{2n}t\ dt-\int_0^{\pi/2}\cos^{2n+2}t\ dt\right]&=\frac{\pi^3}{8}\left[\frac{(2n-1)!!}{(2n)!!}-\frac{(2n+1)!!}{(2n+...
1
vote
1answer
53 views

Does this series $\sum_{n=0}^{\infty}n\left(\frac{4}{5}\right)^n$ converge or diverge?

$\sum_{n=0}^{\infty}n\left(\frac{4}{5}\right)^n$ I'm trying to use the root test and wondering if my steps are correct. First I rewrote the problem: $\frac{n4^n}{5^n}$ Then I did the following: $...
6
votes
3answers
90 views

Puiseux Expansion of Gamma Function about Infinity

In trying to find interesting proofs that Student's T Distribution converges to the Regularized Normal Distribution when $k$ (the number of desgrees of freedom) grows without bounds (i.e. $= \infty$). ...
0
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0answers
31 views

Exercise from David Williams' “Probability with Martingales” page 25

I was stuck at the exercise page 25 and then I found the answer here : Noob Question : Need help to understand : Probability with Martingales : page 25 But there is still one point I don't get, how ...
2
votes
2answers
36 views

Sum of a series (combining divergent ones)

I am reading in a book, without any explanation, the following identity (with $a<b$): $$\sum_{k=0}^{\infty}\left(\frac{1}{k+a+1}-\frac{1}{k+b+1}\right)=\frac{1}{a+1}+\dots +\frac{1}{b}$$ ...
6
votes
0answers
134 views

Generalized Harmonic Number Summation $ \sum_{n=1}^{\infty} \frac{(H_{n}^{(2)})^2}{2^n}$

Prove That $$ \sum_{n=1}^{\infty} \dfrac{(H_{n}^{(2)})^2}{2^n} = \tfrac{1}{360}\pi^4 - \tfrac16\pi^2\ln^22 + \tfrac16\ln^42 + 2\mathrm{Li}_4(\tfrac12) + \zeta(3)\ln2 $$ Notation : $ \...
5
votes
3answers
248 views

Why do $x^{x^{x^{\dots}}}=2$ and $x^{x^{x^{\dots}}}=4$ have the same positive root $\sqrt 2$?

What is $x$ when is satisfies $x^{x^{x^{\dots}}}=2$ ? I am really confused with this; the root is $\sqrt{2}$, but why does the equation $x^{x^{x^{\dots}}}=4$ have the same root?
3
votes
2answers
33 views

Zero-based numbering vs. one-based numbering for sequences

What are the advantages of using one over the other? I mean this in the context of sequences and series. For example, should we let the geometric sequence start from $n=0$ or $n=1$ to get $a_n = a_0r^...
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votes
1answer
86 views

Definite Integral using its McLaurin Series.

I'm trying to solve the next integral, using its series. However, I got stuck in a very dumb way nearly at the end. The infamous: $$\int_{0}^{1} \frac{\text{cosh}(x)-1}{x}dx$$ First, the series of $\...
0
votes
2answers
101 views

Find the integer part of the sum $S=\sum_{k=1}^{80} \frac{1}{\sqrt k} $

Let $$S=\sum_{k=1}^{80} \frac{1}{\sqrt k}.$$Then I would like to obtain $\lfloor S \rfloor$, the integer part of $S$. I am not able to think how to start question .
1
vote
2answers
61 views

Limit of partial sums: $\lim_{n\to\infty} \frac1n\sum_{k=1}^n f(k)=0$ if $\lim_{k\rightarrow\infty}f(k)=0$ [duplicate]

I want to argue that $$\lim_{n\rightarrow\infty} \frac{1}{n}\sum_{k=1}^n f(k)=0~~~~~~~ {\rm if}~~~~~ \lim_{k\rightarrow\infty}f(k)=0.$$ This identity does not seem to hold always, but seems to hold ...
12
votes
1answer
256 views

Can different choices of regulator assign different values to the same divergent series?

Physicists often assign a finite value to a divergent series $\sum_{n=0}^\infty a_n$ via the following regularization scheme: they find a sequence of analytic functions $f_n(z)$ such that $f_n(0) = ...
5
votes
2answers
99 views

Are all infinite sums not divergent? In quantum field theory

I am a physicist interested in physics. In particular this question is related to quantum field theory. I recently came across a derivation of the infinite sum $1+1+1+1+..... $ that produced the ...
0
votes
1answer
29 views

Given an automatic sequence, what monoid endomorphisms fix the corresponding morphic word?

Recall that every automatic sequence forms a morphic word. Given an automatic sequence, how can one construct a monoid endomorphism to define the corresponding morphic word? For example, consider the ...
1
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4answers
69 views

Prove if $0<a<1$ then $\lim_{n\rightarrow\infty}a^{n}=0 $

Good morning i have a proof of this exercise but i don't know if my proof is correct. please review that! Problem: Prove if $0<a<1$ then $\lim_{n\rightarrow\infty}a^{n}=0 $ Proof: Let $...
2
votes
1answer
80 views

The sequence $\{a_k\}_{k=1}^\infty$ is increasing. What about $\{b_k\}_{k=1}^\infty$?

Define the $k^\text{th}$ term of the sequence $\{a_k\}_{k=1}^\infty$ of real numbers by $$a_k := k^2 \int_{-\infty}^\infty e^{-kx^2} \, dx.$$ Then $\{a_k\}_{k=1}^\infty$ is an increasing sequence ...
5
votes
2answers
50 views

Infinite Ordinal Sum

When working with ordinal numbers, would it be correct to say that: $$ \sum_{i=0}^{\infty}1 = \omega$$ Or does this simply not make sense? In the ordinals, does the notation $\sum^\infty_{i=0}$ even ...
1
vote
1answer
42 views

Is $\sum_{n \neq 0}\left(\frac{1+∣a∣}{1+|a-n|}\right)^{100}e^{-n^2}$ bounded independently of $a$?

Fix an $a\in \mathbb{R}$ and consider the sum $$\sum_{n \neq 0}\left(\frac{1+∣a∣}{1+|a-n|}\right)^{100}e^{-n^2}.$$ Is this sum bounded independent of $a$? I think the answer should be yes since for $...
0
votes
1answer
36 views

Suppose a sequence's subsequences have at least one subsubsequence that converges almost surely to $X$. Prove convergence in probability

Probability with Martingales What I tried: 'only if' Suppose a sequence converges in probability to $X$. By $d$ there exists a subsequence that converges almost surelyto $X$. Then by $a$, ...
2
votes
8answers
147 views

Prove if $a>1$ then $\lim_{n\rightarrow\infty}a^{n}=\infty $

Good morning i was thinking about this problem and I make this. I need someone review my exercise and say me if that good or bad. Thank! Problem: Prove if $a>1$ then $\lim_{n\rightarrow\infty}a^{...
1
vote
0answers
44 views

Divergence when spectral radius is greater than one in an iterative map.

Let $(M_n)$ be a convergent sequence of matrices from $\mathbb{R}^p$ to $\mathbb{R}^p$. Each element of the sequence has the same spectral radius $\sigma$, and $\sigma\ge1$. Show that there exist an $...
0
votes
0answers
14 views

Sum of a converging series having Error function with a polynomial

I am struggling to find the sum of the following series: $k\sum\limits_{z=1}^{\infty} \frac{(z+1)^2}{4} . erfc(az)$ where $k$ and $a$ are known parameters and $erfc(x)$ is the complementary error ...
3
votes
2answers
64 views

radius of convergence of $1/(1+z^2)$ about $z=2$ using geometric series approach

I would like to calculate the radius of convergence of $f(z)= 1/(1+z^2)$ about $z=2$ using the geometric series approach. Let me first state that according to a theorem, the radius of convergence ...
1
vote
1answer
46 views

Problem in solving a question related to sequence.

The question is : Prove that the equation $x^{n}+x^{n-1}+...+x-1 = 0$ has exactly one positive root for all $n \in N$ and if $\{b_n\}$ be the sequence of all positive roots of these equations then ...
0
votes
1answer
67 views

$a_{n+1} = (a_n+1)a+(a+1)a_n+2\sqrt{a(a+1)a_n(a_n+1)} \quad (n = 1,2,\ldots).$

Let $a$ be a positive integer and $\{a_n\}$ be defined by $a_0 = 0$ and $$a_{n+1} = (a_n+1)a+(a+1)a_n+2\sqrt{a(a+1)a_n(a_n+1)} \quad (n = 1,2,\ldots).$$ Show that for each positive integer $n$, $a_n$ ...
8
votes
2answers
69 views

Sum to closed form

I need to evaluate the following summation: $$ \sum_{n\in\mathbb{Z}} \frac{-1}{i(2n+1)\pi -\mu} $$ where $n$ is summed over all the integers from $-\infty$ to $\infty$ including 0. Putting this into ...
2
votes
1answer
90 views

Evaluate $\sum_{k=2}^\infty\frac{1}{k^3-1}$

I was considering a specialization of the Cauchy product $$ \left(\sum_{n=1} ^\infty x^n \right) \left(\sum_{n=1}^\infty (-1)^n x^n \right)=\frac{-x^2}{1-x^2},$$ that converges for $0<x<1$. ...
2
votes
3answers
73 views

Does the geometric series have any special result for $r = \phi$?

I am wondering whether partial sums of the geometric series $$S_n = \sum_{k=0}^n r^k = \frac{1 - r^{n+1}}{1-r},$$ and especially, $$S'_n = \sum_{k=\color{red}{1}}^n r^k = r \cdot \frac{1 - r^n}{1-...