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

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2
votes
0answers
37 views

For which $s$ is defined $\frac{1}{\zeta(s)}\sum_{n=1}^\infty\frac{n}{ \left( \zeta(s) \right) ^n}$, where $\zeta(s)$ is the Riemann Zeta function?

If there are no mistakes in my words, if one of the factors $\sum_{n=1}^\infty a_n$ of a convolution product or Cauchy product converges, and the other $\sum_{n=1}^\infty b_n$ corverges absolutely ...
1
vote
1answer
25 views

What is the coefficient and constant term in the following sequence defined recursively?

Let $f_n(x)$ be a sequence of polynomials defined inductively as $f_1(x) = (x - 2)^2$ $f_{n+1}(x) = (f_n(x) - 2)^2$ $; n \ge 1$ Let $a_n$ and $b_n$ respectively denote the constant term and the ...
0
votes
2answers
19 views

bounded but not convergent sequences

I am not sure that if this question has a positive answer...I am looking for a sequence of real numbers $(p_{n})_{n\geq 1}$ such that $-1<\lim _{n}\inf p_{n}\leq \lim_{n}\sup p_{n} <1$ (as ...
0
votes
1answer
27 views

Series proof needed

I have following equations but I do not know the proof. Kindly provide the proof or give me some reference to look into. Here are the equations. 1- ...
-2
votes
0answers
27 views

Why is it that when n ≥ 1 the series is $\le$ 1/4 [closed]

So how is the series $\sum_{n=1}^\infty \frac{1^2 * 3^2 * 5^2 ... (2n-1)^2}{2^2 * 4^2 * 6^2 ... (2n)^2}$ < 1/4 for n $\ge$ 1 is it because the series is divergent outside of the interval of ...
0
votes
0answers
21 views

show function has a uniformly convergent subsequence

alright, so I have this question from my analysis class and I believe I have answered it correctly. I would be grateful if you could read it and give me your thoughts. A sequence $f_n$ of real valued ...
2
votes
1answer
24 views

How to show that all sequentially compact spaces are bounded?

I want to show given a sequentially compact subset $A \subseteq M \implies A$ is bounded. I read this Every sequentially compact set is closed and bounded. but the proof is poorly written and jumpy ...
0
votes
1answer
35 views

Does equality of the sum of two such series imply equality of each term of that series?

Let a(1)< a(2) < ..< a(m) and b(1)< b(2)<..< b(n) be real numbers such that $$\sum_{i=1}^m |a(i)-x| = \sum_{j=1}^n |b(j)-x|$$ for all x belonging to R. Show that m=n and ...
0
votes
0answers
40 views

When adding or subtracting two infinite sums, why is there no issue with “staggering” or arbitrarily manipulating the “alignment” of terms?

I was watching Ramanujan: Making sense of 1+2+3+... = -1/12, where the presenter writes: (I tried to write this out in $\LaTeX$ but couldn't figure out how to do multi-column alignment without ...
0
votes
2answers
576 views

Confused about series and testing for convergence/divergence?

I'm finding it quite difficult to understand the idea of series and limits to test for divergence or convergence. Perhaps more so in finding such a limit. I have the series $$\sum_{n=1}^\infty ...
0
votes
1answer
21 views

Sum of product of convergent series

Suppose $\{a_n\},\ \{c_n\}\subset\mathbb{R}$ satisfies $\lim\limits_{n\to\infty} a_n = a\in \mathbb{R}$, $\lim\limits_{n\to\infty} c_n = 1$. Prove that $$ \lim\limits_{M\to\infty} ...
0
votes
1answer
28 views

Proof of the Strenghtened Limit Comparison Test

I'm studying on my own using Bonar and Khoury's Real Infinite Series. I understand the proof of the "regular" Limit Comparison Test( a link to google books, page 23 ) but the book doesn't provide a ...
0
votes
2answers
53 views

$\sum_{k=1}^n 1/k - \log n$

I got this question : $$a_n = \sum_{k=1}^n \frac 1k - \log n$$ I proved that $\lim a_n $ exist. Now I have to prove: $$ 0<a_n-\lim a_n\le \frac 1n $$ for every $n \in \mathbb N$. I tried ...
4
votes
2answers
138 views

A result on sequences: $x_n\to x$ implies $\frac{x_1+\dots+x_n}n\to x$ without using Stolz-Cesaro

If $x_n \to x$, how might we prove $$\lim_{n \to \infty} \frac{\sum_{i=1}^{n} x_i}{n} = x$$ Of course, one has $\limsup x_n = \liminf x_n = x$, and thus, using the Stolz-Cesaro theorem: $$\liminf ...
1
vote
1answer
18 views

Is this an incorrect error bound value?

In Step 3, they are determining the $(n+1)^{th}$ term. I think the proofreader just added 1, instead of subbing in (n+1). Is that right? I think the correct term should be ...
10
votes
1answer
306 views

Limit of recursive sequence $n^2q_n=1+(n-1)^2q_{n-1}+2(n-2)q_{n-2}$

When looking at this riddle, I came across the following sequence for the frequency of sampled integers between 1 and $n$ in a without replacement/without neighbour sampling: $$q_1=1,\quad ...
0
votes
1answer
21 views

is this an upper bound for the square of a finite series?

I'm trying to show that the derivative of a (long) function is negative. after a lot of simplifying, i need the following inequality to hold: $\displaystyle \Big ( \sum_{i=1}^k n_i \Big )^2 < ...
-1
votes
2answers
42 views

$\sum \frac{b_j}{j^2}$ converge or diverge

if $b_j>0$ and $\sum b_j$ convege does $\sum \frac{b_j}{j^2}$ converge or diverge I think it converges because $\sum \frac{b_j}{j^2} < \sum b_j$ Thus by comparison test converges.
114
votes
1answer
4k views

Identification of a curious function

During computation of some Shapley values (details below), I encountered the following function: $$ f\left(\sum_{k \geq 0} 2^{-p_k}\right) = \sum_{k \geq 0} \frac{1}{(p_k+1)\binom{p_k}{k}}, $$ where ...
0
votes
1answer
42 views

Radius of Convergence of $\sum_{n = 0}^{\infty} \frac{(-1)^nn!x^n}{n^n}$

I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide explanations, and I'd really, really appreciate it if ...
0
votes
1answer
51 views

If $A_n = \sum_{k = 1}^n \dfrac{k^6}{2^k}$, find $\lim_{n \to \infty} A_n$

Set $$A_n = \sum_{k = 1}^n \dfrac{k^6}{2^k}.$$ Find $\displaystyle \lim_{n \to \infty} A_n$. I tried solving this using a reduction method. That is, reducing the above series to an ...
0
votes
1answer
26 views

How would I show $|x| \le 1$ given the equation for $x$ the expression in the equation?

The expression is $x = \sin(\theta /2)$. I am asking how would I show that $\sin(\theta/2)\le1$ based on the expression? I already know that the biggest $\sin$ will ever get is $[-1, 1]$ which is the ...
0
votes
1answer
38 views

Investigate convergence of $\sum_{n=2}^\infty \frac{1}{n(n-1)}$

Investigate convergence of: $\sum_{n=2}^\infty \frac{1}{n(n-1)}$ Now I know by $p$-series test this summation converges however, is there a way to prove that this series converges by some ...
1
vote
2answers
25 views

Solving inequality with a recursive formula without its closed form?

I have the following problem: $$a_{1}=1$$ $$a_{2}=3$$ $$a_{n+2}=a_{n+1}+5a_{n}$$ I have to prove this inequality: $$a_{n}<1+3^{n-1}$$ So my question, is there a way to solve this inequality without ...
12
votes
6answers
16k views

Why does the harmonic series diverge but the p-harmonic series converge

I am struggling understanding intuitively why the harmonic series diverges but the p-harmonic series converges. I know there are methods and applications to prove convergence, but I am only having ...
3
votes
3answers
562 views

interval of convergence of $\sum n \exp (-x \sqrt n)$

$$\sum^{\infty}_{n=1} n \exp (-x \sqrt n)$$ How to find the interval of convergence? Obviously, 0 is not in the interval because the series becomes divergent. could you help me?
34
votes
2answers
2k views

Is $ \sum\limits_{n=1}^\infty \frac{|\sin n|^n}n$ convergent?

Is the series $$ \sum_{n=1}^\infty \frac{|\sin n|^n}n\tag{1}$$ convergent? If one want to use Abel's test, is $$ \sum_{n=1}^\infty |\sin n|^n\tag{2}$$ convergent? Thank you very much
9
votes
2answers
550 views

Is there an analytic function satisfying $\,\,f\big(\!\frac 1 n\!\big)=\frac 1 {\sqrt{n}}$ for all $n\in\mathbb N$?

Is there a function which is analytic in an open neighbourhood of $z=0$ and satisfies $$ f\left(\!\dfrac 1 n\!\right)=\dfrac 1 {\sqrt{n}}, $$ for all natural numbers $n$? I guess this problem ...
-3
votes
0answers
66 views

What is the similarity between the numbers 85, 17, 19, 4 and 2? [closed]

What is the similarity between the numbers 85, 17, 19, 4 and 2? this a brain game question airing at a local radio station with a good jackpot on it, its either a sequence, a date in historical events ...
2
votes
1answer
2k views

What is the definition of a geometric progression?

If the first term in our geometric progression (GP) is $k$, and the common ratio is 0, then our sequence is $\{k, 0, 0, 0, 0,\ldots\}$. Is there anything wrong with this statement? So, is $\{0, 0, ...
3
votes
2answers
47 views

Determine wether $\sum_{n=1}^\infty \frac{3n-1}{(n+1)^3}$ converges or diverges.

Determine wether the following function converges or diverges by comparison test: $\sum_{n=1}^\infty \frac{3n-1}{(n+1)^3}$ Upon inspection I can clearly see that the series converges. However I am ...
0
votes
1answer
34 views

Is there a general approach to find an explicit formula to a recursive sequence?

I've been doing a bunch of exercises where I need to find the explicit formula for a given sequence... $$a_{n+2} = 6 a_{n+1} - 9 a_{n}$$ The first were easy with $a_{0} = 1$ and $a_{1} = 3$ i.e.. But ...
3
votes
3answers
126 views

How is the last “=” true?

How can the last equality be true? $$ G(u)=\frac{g}{(1+u)^g-1}-\frac1u=\frac{g}{gu + \cdots + u^g} - \frac{1}{u}=\sum_{k=0}^\infty c_ku^k $$
2
votes
0answers
38 views

Tighter upper bounds with ratios of powers of norms

This question arises in concentration or sparsity measures for finite sequences. Given $x\in \mathbb{R}^K$ and $1 \le r < s$, I try to find a tight upper bound for $$\psi_{r,s}(x) = \frac{\sum_1^K ...
6
votes
1answer
54 views

Show $x_n := \frac{1-p^{n+1}}{1-p^n} \frac{n}{n+1}$ is increasing in n for $p \in (0,1)$

I need to show that $x_n := \frac{1-p^{n+1}}{1-p^n} \frac{n}{n+1}$ is increasing in $n$ for $p \in (0,1)$. My attempts have involved trying to show $\begin{eqnarray*} \frac{1-p^{n+2}}{1-p^{n+1}} ...
1
vote
1answer
30 views

Proof of a theorem regarding subsequences and convergence

I am attempting to understand the following theorem and it's proof as outlined in my textbook for Real Analysis. Theorem: Let $(s_n)$ be a sequence. If t is in $\mathbb{R}$, then there is a ...
4
votes
1answer
91 views

Convergence and value of infinite product $\prod^{\infty}_{n=1} n \sin \left( \frac{1}{n} \right)$?

Since the limit $\frac{\sin(x)}{x}=1$ for $x \rightarrow 0$, I wondered about the infinite product: $$\prod^{\infty}_{n=1} n \sin \left( \frac{1}{n} \right)=\sin(1) \cdot 2 \sin\left( \frac{1}{2} ...
0
votes
1answer
22 views

Show the following series converges uniformly using Weierstrass M Test

I'm trying to show that the following series converge uniformly by using the Weierstrass $M$ Test. $$ \sum ^{\infty}_{j=0}z^{n},\ \ \ 0\leq \left | z \right |< R,\ \ \ R<1 $$ and $$ \sum ...
3
votes
2answers
35 views

$\lim x_n = a$, $\lim \frac{x_n}{y_n}=b$ then $\lim y_n = \frac{a}{b}$

I must prove: $\lim x_n = a$, $\lim \frac{x_n}{y_n}=b$ then $\lim y_n = \frac{a}{b}$ Well, I know that $$\lim x_n = a \implies |x_n-a|<\epsilon$$ $$\lim \frac{x_n}{y_n} = b \implies ...
-2
votes
0answers
22 views

Finding the nth term of a geometric series [closed]

Please help me find the nth term of the series: $$\frac{2}{5}-\frac{6}{5^2}+\frac{10}{5^3}-\frac{14}{5^4}+\cdots$$
0
votes
1answer
26 views

Show ${f_n(x)}_ {n=1, \cdots, \infty}$ converges to $0$ uniformly on $(0,1)$

If $f_n(x) = \dfrac{x}{1+nx}$, show that ${f_n(x)}_{n=1, \cdots, \infty}$ converges uniformly to $0$ on $(0,1)$. Here is what I have so far: Let $\epsilon>0$ be given. Pick any $n \in ...
1
vote
1answer
22 views

$a_n, b_n$ bounded, $a_n+b_n=1$,$z_n\to a$ and $t_n\to a$, then $(a_nz_n+b_nt_n)\to a$

I must show that if $a_n, b_n$ are bounded such that $a_n+b_n=1$, and if $z_n\to a$ and $t_n\to a$, then $(a_nz_n+b_nt_n)\to a$ My idea was: $$(a_n+b_n)(z_n+t_n) = a_nz_n+a_nt_n+b_nz_n+b_nt_n$$ I ...
1
vote
1answer
15 views

What is the Laurent expansion of f(z)=1/(z-3)?

What is the Laurent expansion of f(z)=1/(z-3)? In the region, ㅣZ-3ㅣ>0 ? I just computed the Laurent expansion in the region ㅣZㅣ>3 by dividing the denominator by 1/z and making it as a geometric ...
1
vote
1answer
24 views

Write the series using sigma notation: $f(x)= 1 - (x^2)/2! + (x^4)/4! - (x^6)/6! +\cdots$

$$f(x)= 1 - (x^2)/2! + (x^4)/4! - (x^6)/6! + \cdots$$ I don't know how to get the signs to work like negative, then positive. I have tried to make it like the following: $(-1)^{n-1} \frac{x^{2n-1}}{ ...
1
vote
3answers
56 views

$\lim a_n = L \implies \lim a_n^2 = L^2$

I have to prove the following: $$\lim a_n = L \implies \lim a_n^2 = L^2$$ I know that $\lim a_n = L \implies \forall\epsilon>0 \ \exists n_0$ such that $n>n_o\implies |a_n-L|<\epsilon$ I ...
-7
votes
0answers
24 views

series convergence help using tests [closed]

Use any theorems or properties of series. This was a question on my homework and I received 0 points. I need help with the entire question. I originally tried to compare part (a) to the ...
0
votes
1answer
33 views

How to prove $\sum n/3^n$ converges without ratio test?

The only tests my class has learned so far and is allowed to use are: Divergence, Integral, Comparison, Geometric/Harmonic/Telescopic. I have proved that the series converges, via a ratio test, but my ...
0
votes
1answer
28 views

Solve for bound of $\sigma(n)$ from harmonic series.

I am given the harmonic series, $H_n$. How can I show that for $n\geq 1$, $$\sigma(n)\leq H_n+e^{H_n}\ln{H_n}$$ By the way, if you're not familiar with it, $\sigma(n)$ is the sum of all positive ...
0
votes
0answers
18 views

limits and convergence of sequences complex

For the following sequence discuss its limits and whether the convergence is uniform, in the region $\alpha \leq \left | z \right |\leq \beta $, for finite $\alpha$,$\beta >0$. $$\left \{ ...
3
votes
2answers
44 views

Sum of the series $\frac{1}{(1-x)(1-x^3)}+\frac{x^2}{(1-x^3)(1-x^5)}+\frac{x^4}{(1-x^5)(1-x^7)}+…$

If $|x|<1$, find the sum of infinite terms of following series: $$\frac{1}{(1-x)(1-x^3)}+\frac{x^2}{(1-x^3)(1-x^5)}+\frac{x^4}{(1-x^5)(1-x^7)}+....$$ Could someone give me hint to solve this ...