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

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-1
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1answer
19 views

The sum of all the odd numbers to infinity

We have this sequence: S1: 1+2+3+4+5+6.. (to infinity) It has been demonstrated, that S1 = -1/12. Now, what happens if i multiply by a factor of 2? S2: 2+4+6+8+10+12.... (to infinity). I have ...
2
votes
3answers
51 views

For the series $S = 1+ \frac{1}{(1+3)}(1+2)^2+\frac{1}{(1+3+5)}(1+2+3)^2$…

Problem : For the series $$S = 1+ \frac{1}{(1+3)}(1+2)^2+\frac{1}{(1+3+5)}(1+2+3)^2+\frac{1}{(1+3+5+7)}(1+2+3+4)^2+\cdots $$ Find the nth term of the series. We know that nth can term of the ...
0
votes
2answers
37 views

How to properly state as to why $\sum_{n=1}^{\infty}\frac{\sqrt{n^3+2}}{n^4+3n^2+1}$ converges.

So I know that $\sum_{n=1}^{\infty}\frac{\sqrt{n^3+2}}{n^4+3n^2+1}$ converges, because the highest power in the numerator is $n^\frac{3}{2}$ and the highest power in the numerator is $n^4$, so I have ...
0
votes
1answer
11 views

Cauchy sequence of vectors when dotted with another vector gives a Cauchy sequence of scalars?

My question is related to vector spaces with an inner product defined (the space is not necessarily complete i.e. not a Hilbert Space) So imagine I have a Cauchy sequence of vectors ...
1
vote
2answers
39 views

How to determine whether $\sum_{n=1}^{\infty}\ln\left(\frac{n+2}{n+1}\right)$ converges or diverges.

I am trying to find whether $\sum_{n=1}^{\infty}\ln\left(\frac{n+2}{n+1}\right)$ converges or diverges. I used the limit test, and it comes out as inconclusive since ...
3
votes
0answers
19 views

Modes of convergence in infinite direct sums of $L^{2}$ spaces

It is known that if a sequence of random variables converges in norm then there exists a subsequence which converges almost surely. That is: let $\left(X_{n}\right)_{n\in\mathbb{N}}\subseteq ...
0
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1answer
29 views

$\{p_{n}\}$ is a sequence of real numbers. Prove $\limsup$ $\{p_{n}\} < \infty$ if and only if $\{p_{n}\}$ is bounded above.

I have done the following. $\Leftarrow$ $\limsup$ $\{p_{n}\}$ is the set of suprema of all the subsequential limit points of $\{p_{n}\}$. So, if it were not finite, then, given any $M\in N$, ...
0
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1answer
29 views

Calculus sequences And series

Find the values of $x$ for which the series $\sum_o^\infty \frac {(x+3)^n}{2^n}$ converges. I took it as $(\frac {x+3}2)^n$ then used the rule of summation of $r^n= \frac 1{1-r}$ then found ...
1
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0answers
19 views

Name for hypergeometric-like sum.

Consider the sum $F(a_1,\cdots,a_r; z)\sum_{m=0}^{\infty}\frac{(a_1+m)(a_2+m)\cdots(a_r+m)}{(a_1)(a_2)\cdots (a_r)}\frac{z^m}{m!},$ where $a_i$ for $i=1,2,\cdots,r$ is an increasing sequence of ...
0
votes
0answers
35 views

Can this convergent series be generalised?

A friend of mine gave this question,I have no idea how to even start generalising the nth term of the series so that I can summify it to n tending to infinity. $$\frac{1}{(1!)} ...
2
votes
1answer
25 views

Making these series equations equal?

I was recently attempting to prove the formulae which calculate the sum of arithmetic sequences where the difference between each term is just 1. I arrived at this formula first, which calculates the ...
0
votes
2answers
24 views

Prove that a sequence of recursive functions $\,f_n(x)$ cannot converge pointwise to $\,f(x)$ on $[0,1]$

Given a recursive sequence $\,f_n(x) :[0,1] \to \mathbb R$, $x \in [0,1]$, where $$\begin{align*} f_1(x) &= x, \\[6pt] f_n(x) &= \frac{2x\,f_{n-1}(x)}{n!} \end{align*}$$ I have proven that the ...
2
votes
3answers
56 views

Convergence of $\frac{\ln(n)}{n^2}$

During a Calc exam, I needed to state whether $$\sum_{n=1}^{∞} \frac{\ln n}{n^2}$$ converged or diverged. I was going to try a strict comparison test (with $\sum_{n=3}^{∞} \frac{\ln n}{n^2}$ to ...
0
votes
1answer
15 views

Limit of sequence involving a product

This question is related to a post that was deleted. I want to calculate the following limit $$\lim_{n \to ...
0
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4answers
45 views

An infinite series question. [on hold]

Ok, so we have an infinite sequence: S1 = 6+14+22+30+38.. Now, there is another infinite sequence, S2 = 1+2+3+4+5+6+7.. We know that the first sequence's nth term = (8n-2 ) so surely there must be a ...
0
votes
0answers
19 views

Rearranging a sequence to minimize a series function on its subsequences

Given an arbitrary nonempty sequence of integers $Q$ with $p$ elements: Let $R$ be a rearrangement of $Q$. Let $A$ be the solution set of $1\leq n\lt m\leq p$, where $n,m\in \mathbb{Z}$. $F(n,m) = ...
0
votes
1answer
18 views

How to prove that f(x)=1/|x-t| is continuous but not bounded?

Suppose S is not closed: there is a point t in R, t not in S, such that a sequence in S converges to t. Show that the function f: S-> R, defined by f(x) = 1/|x -t|, is continuous but not bounded.
1
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0answers
22 views

Showing a series is not the fourier series of a riemann integrable function.

I want to show that the series $\sum_1^\infty \frac{sin(nx)}{\sqrt{n}}$ is not the Fourier series of a Riemann integrable function on $[-\pi,\pi]$. I was going to do this by showing that the partial ...
3
votes
6answers
70 views

Prove that the sequenze $b_n=\left(1+\frac{1}{n}\right)^{n+1}$ is decreasing [duplicate]

Prove that the sequence $$b_n=\left(1+\frac{1}{n}\right)^{n+1}$$ Is decreasing. I have calculated $b_n/b_{n-1}$ but it is obtain: $$\left(1-\frac{1}{n^2}\right)^n \left(1+\frac{1}{n}\right)^n$$ But I ...
-3
votes
3answers
41 views

Convergence series $\sum_{n=1}^\infty n\sin\left(\frac{1}{n^2}\right)$ [on hold]

You can help me to show if the following series converges or diverge. $$\sum_{n=1}^\infty n\sin\left(\frac{1}{n^2}\right)$$
2
votes
2answers
27 views

Real convergent sequences

Let $(a_n)$ be a bounded sequence for all $n$ such that $ \displaystyle a_n \geq \frac{1}{2} (a_{n-1}+a_{n+1})$ for $n\geq 2$. Show that $(a_n)$ converges. I think I cannot use any convergence tests ...
0
votes
0answers
16 views

Paperfolding Constant

In the Wikipedia article about the regular paperfolding sequence, it says (more or less quoted): Taking $G(t_n;x) = G(t_n;x^2) + \sum_{n=0}^{\infty}x^{4n+1} = > G(t_n;x^2) + \frac{x}{1-x^4}$ ...
0
votes
1answer
37 views

prove if is converges or diverges sequence

find if it's converges or diverges, if converges find the limit: $$\frac{(-1)^n n+1}{n^2+1}.$$ My proof: divided by n^2 so you have $\frac{(-1)^n(1/n+1/n^2)}{1+1/n^2}$ if I take the limit $n\to ...
4
votes
1answer
86 views

L'Hopital's Rule, Factorials, and Derivatives

I have the following limit $\displaystyle\lim_{n\to\infty}\frac{e^n}{n!}$. Now if I try to solve this using this using L'hopital's rule, I won't be able to since I can't take the derivative of $n!$. ...
0
votes
1answer
17 views

Calculate duration of task

Say I have some task to process 100 days of data, and it takes 5 hrs to process a day. But each day that it takes to process it a new day of data comes in. So for the initial set of data it takes: 5 ...
0
votes
1answer
21 views

Upper and Lower Limits of a Sequence

If we partition a sequence into a finite number of subsequences then the upper and lower limit of the sequence are equal to the maximum upper limit and minimum lower limit of the subsequences. Has ...
0
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0answers
17 views

Any suggestions to find a non recursive formula for this sequence?

I have these elements in a set $R$, $$\{r_i\ /\ i=1,2,3,\dots,2n \}\in R$$ And we define: $$a(1)=r_1+r_2$$ $$a(2)=\frac{(r_1+r_2)\cdot r_3}{(r_1+r_2)+ r_3}+r_4$$ ...
1
vote
1answer
15 views

Summation of arithmetic-geometric series of higher order

There is a closed formula for the summation of arithmetic-geometric series: $\sum_{x=1}^{+\infty }(ax+b)r^x=\frac{(a+b)r-br^2}{(1-r)^2}$ when $-1<r<1$ But consider: $\sum_{x=1}^{+\infty ...
0
votes
0answers
11 views

Expressing one generating function like combination of another generating functions.

Let A (t), B (t) and C (t) - generating functions for sequences $a_0, a_1, a_2,\dots; b_0, b_1, b_2,\dots and\ c_0, c_1, c_2,\dots$ Express C (t) through A (t) and B (t), if ...
1
vote
1answer
53 views

How to prove this sequence is convergent?

Suppose that the series $\sum_{k=1}^\infty{a_k}$ converges. Prove that $$\lim_{n→\infty}\frac{1}{n}\sum_{k=1}^{n}ka_k=0$$ I tried to use the definition of convergence of $\sum a_k$ ...
0
votes
0answers
26 views

If $a_i$ is an increasing sequence then which of the following is convergent?

If $a_i$ is an increasing sequence then which of the following is convergent? $\sum_{i=1}^{n}a_i$ $\sum_{i=1}^{n}\frac{a_{i+1}}{a_i}$ $\sum_{i=1}^{n}\frac{1}{a_i^2}$ $\sum_{i=1}^{n}\sqrt{a_i}$ I ...
4
votes
3answers
37 views

Calculating a limit with infinitely many terms

I've encountered this limit : $$\lim_{n\to\infty} \frac1n \left(\sin\left(\frac{\pi}{n}\right) + \sin\left(\frac{2\pi}{n}\right)+\cdots+\sin{\frac{(n-1)\pi}{n}}\right)$$ Wolfram gives the value: ...
0
votes
1answer
40 views

Arctan(f(x)) is almost the same as Erf(f(x)) for many f(x). Is the just coincidence or is there a reason?

For example: Arctan(x) is almost Erf(x) (subtle differences in absolute value and curve) Arctan(x^50) is almost Erf(x^50) (difference in absolute value) and many others, so we can conclude: ...
0
votes
1answer
10 views

Find the convergence radius for $ \sum_{k=0}^{\infty} \frac{k(z+i)^{2k}}{2^k} $

What's the convergence radius for $$ \sum_{k=0}^{\infty} \frac{k(z+i)^{2k}}{2^k} $$ I using the root criterium that says that the serie convergence if the limit is 0. $$ lim_{k \rightarrow \infty} ...
1
vote
2answers
32 views

convergence of $\sum_n x_n$ versus $\sum_n f(x_n)$ for differentiable $f$

Suppose $f(0) = 0$, $f$ is differentiable at 0, $f'(0) \ne 0$, and $x_n \rightarrow 0$. What can we say about (i) the convergence of $\sum_n x_n$ versus (ii) the convergence of $\sum_n f(x_n)$? It ...
-1
votes
2answers
31 views

How to prove that $\sum_{i=0}^h2^i=2^{h+1}-1$ [on hold]

How do I prove the following relationship? $$\sum_{i=0}^h2^i=2^{h+1}-1$$
2
votes
1answer
17 views

Which of the following are true about sequences?

If $(x_n)$ is a sequence of real numbers such that for every $n$ we have $0<x_n<\frac{1}{n}$ then which of the following is true? $1.\lim_{n\to\infty}x_n=0$ $2.$If $f$ is continuous function ...
0
votes
2answers
13 views

convergence of a series of function

I have to find the set of pointwise and uniform convergence of this series: $\sum x(1-x)^n$. The set of pointwise convergence is $[0,2)$. But for the uniform convergence what can I do?
3
votes
1answer
137 views

Find the limit of recursive sequence, if it exists (Analysis, calculus)

My goal is to to test this recursive sequence if it's convergent and if yes, find the limit. $$a_1=3,\:a_{n+1}=\frac{7+3a_n}{3+a_n}$$ I know how to do this with normal sequences, but this is the ...
0
votes
1answer
27 views

A question about infinitie series and pi

This is the sequence that can be used to find an exact value of pi 4/1−4/3+4/5−4/7+4/9−4/11…..(to infinity) = 𝜋 Or (1/1−1/3+1/5−1/7+1/9−1/11….. (to infinity) )= 𝜋/4 Given that we have this ...
0
votes
1answer
20 views

Absolute Convergene of the Product Series

Theorem Suppose (a) $\sum_{n=0}^{\infty}a_n$ converges absolutely, (b) $\sum_{n=0}^{\infty}a_n=A$, (c)$\sum_{n=0}^{\infty}b_n=B$, (d)$c_n=\sum_{k=0}^{n}a_kb_{n-k}$ $(n=0,1,2,\dots)$. Then ...
-1
votes
2answers
98 views

What is the formula to generate this number sequence : 1 , 7 , 14, 30

What is the formula to generate this number sequence : 1 , 7 , 14, 30 I'm sure this is very simple for you guys. But it's got me alittle stuck. Thanks To clarify, I'm not an advanced maths student. ...
0
votes
0answers
20 views

Prove the combination rule for non-negative series by using comparison test

The combination rule for non-negative series is as follows: The question is, how can I prove combination rule for non-negative series by using the comparison test, namely:
0
votes
1answer
28 views

summation of series by telescoping series method (feedback needed)

i am stuck i did the first part by cancelling out terms since its a telescoping series. But I do not know how I can proceed any further . Please help. I am not sure of whatever i have done so far. so ...
2
votes
3answers
126 views

Find if this series converges and if so find its value

I need help I cant understand how we can solve this. I am confused when the log came in. I listed the first few terms but i do not know how to proceed further. all I know is that the sequence is ...
0
votes
0answers
37 views

Study the series $\sum_{n=1}^{\infty} \left|\frac{1}{n^{\cos\theta+ i \sin \theta}}\right|$

Study the character of the series $$\sum_{n=1}^{\infty} \left|\frac{1}{n^{\cos\theta+ i \sin \theta}}\right|$$ With i is the immaginary unit, $\theta$ is a real angle. My answer is that the series ...
-1
votes
2answers
31 views

Theorem 3.29 in Baby Rudin

Theorem 3.29 in Walter Rudin's Principles of Mathematical Analysis, 3rd ed., states that If $p>1$, then the series $$\sum_{n=2}^\infty \frac{1}{n (\log n)^p} $$ converges; if $p \leq 1$, the ...
2
votes
1answer
35 views

Definition of the limit of a sequence

I'm looking over the following definition of convergent limits: A sequence $(x_n)$ in $\mathbb{R}$ is said to converge to $x \in \mathbb{R}$, or x is said to be a limit of $(x_n)$, if for every ...
1
vote
2answers
71 views

What branch of analysis deals most with sequences and series?

I'm really interested in sequences and series (mainly series). What kind of math branch should I look more into? I understand that sequences and series mostly point toward analysis, but what ...
1
vote
1answer
26 views

Recognising that $\sum_{n=0}^\infty \frac{a^2-b^2(2n+1)^2}{(a^2+b^2(2n+1)^2)^2}=-\frac{\pi^2\mathrm{sech}^2\left(\frac{a\pi}{2b}\right)}{8b^2}$

So I know from Mathematica that: $$\sum_{n=0}^\infty \frac{a^2-b^2(2n+1)^2}{(a^2+b^2(2n+1)^2)^2}=-\frac{\pi^2\mathrm{sech}^2\left(\frac{a\pi}{2b}\right)}{8b^2}$$ I am wondering how someone could ...