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

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0answers
3 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
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5answers
46 views

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

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 ...
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3answers
32 views

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

You can help me to show if the following series converges or diverge. $$\sum_{n=1}^\infty n\sin\left(\frac{1}{n^2}\right)$$
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2answers
17 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 ...
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0answers
10 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
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1answer
34 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
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1answer
72 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
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1answer
15 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
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1answer
19 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 ...
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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: $$a(1)=r_1+r_2$$ $$a(2)=\frac{(r_1+r_2)\cdot r_3}{(r_1+r_2)+ r_3}+r_4$$ ...
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1answer
14 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 ...
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0answers
9 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
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1answer
44 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
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0answers
24 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
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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
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1answer
38 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
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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} ...
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2answers
31 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 ...
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2answers
30 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
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1answer
16 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
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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?
2
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1answer
92 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
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1answer
25 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
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1answer
18 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 ...
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2answers
94 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. ...
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0answers
16 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:
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1answer
25 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
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3answers
121 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
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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 ...
0
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2answers
30 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
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1answer
34 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 ...
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2answers
68 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
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1answer
25 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 ...
0
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2answers
33 views

Prove that sequences $\frac{a_n}{b_n} = 0$

Let $(a_n)$ and $(b_n)$ be positive real sequences such that $\lim \limits_{n \to \infty} \dfrac{a_n}{b_n} = 0$ and $(b_n)$ is bounded. Prove that $\lim \limits_{n \to \infty} a_n=0$. Proof: ...
0
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1answer
31 views

Radius and Interval of Convergence of $\sum\limits_{n=1}^{\infty}\dfrac{x^n}{2n-1}$

This is my first time finding the radius and interval of convergence of a series, so please bear with me. I would like to find the radius and interval of convergence of ...
2
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1answer
27 views

Series and comparison test

If $a_n>0$ and $\sum a_n$ diverges, what can be said about $\displaystyle \sum \frac{a_n}{1+na_n}$? I cannot prove that it is convergent or divergent. I think it is convergent for some examples ...
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1answer
34 views

Test whether $\sum_{n=1}^{\infty}\frac{\ln{n}}{n}$ converges or diverges

I am trying to solve this using an integral test, but I am unsure whether or not this is correct. Let $f:[2,\infty)\to\mathbb{R}$ be defined by $f(t)=\frac{\ln{(t)}}{t} >0\ \forall t\geq2$. Now ...
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2answers
52 views

Limit of a headache-giving series [on hold]

I found this problem somewhere which says to find out the limit of this series, in order to prove that the limit is somewhere outside $\mathbb{Q}$. $$ x_n = \sum_{k=0}^n 2^{-k^2-k}\;,\quad \forall n ...
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0answers
24 views

Recursive sequence with a formula for a part of its criteria

I have the next recursive sequence which firts terms are $$2,\ \frac{3}{2},\ \frac{10}{7},\ \frac{17}{12},\ \frac{112}{89}$$ I need to express it has a general form for the $n$th element, I can't make ...
0
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2answers
17 views

Limits of sequences and series

I'm currently learning about the limits of a sequence based on the following definition: A sequence $x_n$ is said to converge to $x \in \mathbb{R}$, if for every $\epsilon > 0 $ there exists a $K ...
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0answers
25 views

Closed form of $\sum_{k=1}^{\infty }\left(\psi_1(k)\right)^n$

Inspired by answers to this question, for which $n$ values could we specify a closed-form of $$S(n)=\sum_{k=1}^{\infty }\left(\psi_1(k)\right)^n\,?$$ Here $\psi_1$ is the trigamma function, and ...
0
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1answer
24 views

Radius of convergence: $\sum_{k=1}^\infty \frac{x^{2k-1}}{2k-1}$

It is asked to find the radius of convergence of the series $$\sum_{k=1}^\infty \frac{x^{2k-1}}{2k-1}$$ i.e, to find the values of x such that this series converges. Clearly, I could directly apply ...
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2answers
49 views

Why $\cos(n^2x) \not\to 0$ for any real $x$?

I'd like to show, as simply as possible, that $\sum_{n=1}^\infty\cos(n^2 x)$ diverges for every real $x$. (I know how to prove it for rational $x$. For irrational $x$, I don't know if $(n^2 x)$ is ...
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2answers
20 views

Determine the interval of convergence of series $\sum_{n=1}^{\infty } \frac{1}{\cos ^2(n \cdot x)+\sqrt{n}}$

So, hey. I was sincerely trying to find it by myself with Weierstrass M-test, but failed occasionally, because I ended up with $\sum_{n=1}^{\infty } \frac{1}{\sqrt{n}}$,which is a divergent series. ...
1
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0answers
17 views

Alternating series involving sums of k-primes

As an exercise, if $p_k$ are positive integers composed of k primes including repetition and $\pi_k(n)$ the number of $p_k$ not exceeding n can we show that for the alternating series of sums of ...
1
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1answer
36 views

Convergence of $\sum_{n=1}^{\infty} \frac{1}{n^z}$

Let us consider $z\in \mathbb C$; what is the condition on modulus of z in order that $$\sum_{n=1}^{\infty} \frac{1}{n^z}$$ the series (zeta function?) converges? For example, if $|z|=1$, the series ...
1
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2answers
47 views

how to solve the following without a calculator

Is their a method for the required question except brute force ? (here $11$ terms gets multiplied as followed) ...
0
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2answers
21 views

Is there a mistake in this page on asymptotic expansions?

I think there is an error in section 4.3 of this page - http://aofa.cs.princeton.edu/40asymptotic/ The author says that by taking $x = -\frac{1}{N}$ in the geometric series $\frac{1}{1-x} = 1 + x + ...
0
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3answers
40 views

How can I determine if the sequence $\frac{n^{2} + 3^{n}}{n^{4} + 2^{n}}$ converges or diverges?

Determine whether the sequence $\frac{n^{2} + 3^{n}}{n^{4} + 2^{n}}$ converges. If it converges find the limit and if it diverges determine whether it has an infinite limit. Proof: let $a_{n} = ...
0
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0answers
24 views

Calculating the power series expansion about pi/2 of g(z)=tan[z/2]

Calculating the power series expansion about pi/2 of g(z)=tan[z/2]. Now calculate the expansion about 0. I'm having trouble doing this. I'm not even sure which is the best way to approach it, for ...