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

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0
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2answers
21 views

Geometric progression

In a G.P. the sum of the first and last terms is equal to 66. the product of the second and second last term is 128. What is the first term. Options are- a) 64 b)64 or 2 c)2 or 32 d) none of the above ...
2
votes
0answers
42 views

Replacing $q^2$ by $q$

I have a rather strange question. Suppose we are given a formal power series $$S(q^2) = \sum_{n = 0}^\infty a_n q^{2n}.$$ I wish to replace $q^2$ by $q$. This implies that $S(q) = \sum_{n = 0}^\infty ...
1
vote
4answers
67 views

Limit and supremum conceptual question

Is it true that for a monotone increasing sequence, the limit of the sequence must be its supremum, but the supremum of the sequence might not be its limit? Else what is the relationship between ...
0
votes
2answers
25 views

Show $\sum_{k=2}^{2n+1}\frac{2}{k^2-1}= \frac{3}{2}-\frac{1}{2n+1}-\frac{1}{2n+2}$

Show that $$\sum_{k=2}^{2n+1}\frac{2}{k^2-1}= \frac{3}{2}-\frac{1}{2n+1}-\frac{1}{2n+2}$$ with the hint, "Write out the first six and last two terms. Then group them in pairs of two." Additionally, ...
1
vote
2answers
45 views

Bounding $\sum_{n=n_1}^\infty x^n (n+1)^2$

I need to upperbound the sum $$\sum_{n=n_1}^\infty x^n (n+1)^2$$ where $0<x<1$ is a parameter. I know it can be done starting from $$\sum_{n=n_1}^\infty x^n (n+1)^2\le \sum_{n=0}^\infty x^n ...
1
vote
1answer
35 views

Find a general formula for x_k

The sequence $x_k$... is defined by $x_0 = 0, x_1 = 2$, and $x_{k+2} = 6x_{k+1}−13x_k$ for $k≥0$. Find a general formula for $x_k$. I actually came here because I found a solution on here for a ...
3
votes
1answer
171 views

What is the 5000th happy prime number?

Im writing a program that finds the Nth happy prime number. I think it works, but to double check I want to compare what it returns for the 5000th happy prime number. The problem is, I dont know where ...
0
votes
0answers
41 views

Superlinearly convergent

A sequence $\{p_n\}$ is said to be superlinearly convergent to $p$ if $$\lim_{n\to \infty}{\frac{|p_{n+1}-p|}{|p_n-p|}}=0$$ a. Show that if $p_n\to p$ of order $\alpha$ for $\alpha>1$, then ...
2
votes
1answer
47 views

$(\ell_p,\|.\|_{\infty})$ Banach or separable

Is $(\ell_p,\|.\|_{\infty})$ for $1\leq p<\infty$ a Banach or separable space? is there any fast way of proving it without checking separability or completeness with the usual way?
0
votes
1answer
48 views

Maclaurin Series multiplying in a constant

So I understand how to set up this series but I'm just confused on the last part so the question is find the maclaurin series for the following: $$f(x) = 15x \cos \left( \frac{1}{14}x^2 \right)$$ so ...
3
votes
1answer
62 views

Alternating series limit question [closed]

Suppose $b_n > 0$ for all $n\geq1$ and $$\lim_{n\to\infty}n\left(\frac{b_n}{b_{n+1}}-1\right)>0,$$ show that the alternating series $\sum_{n=1}^\infty(-1)^n b_n$ converges. Any hints?
2
votes
3answers
31 views

Demonstrate that the sequence is decreasing

$$\left( 1-\frac{1}{n+1}\right)^n$$ How do I demonstrate that the sequence decreases? I tried using the newton binomial but I end up with a terribly complicated expression.
8
votes
4answers
332 views

Find the infinite sum of the series $\sum_{n=1}^\infty \frac{1}{n^2 +1}$

This is a homework question whereby I am supposed to evaluate: $$\sum_{n=1}^\infty \frac{1}{n^2 +1}$$ Wolfram Alpha outputs the answer as $$\frac{1}{2}(\pi \coth(\pi) - 1)$$ But I have no idea ...
0
votes
1answer
73 views

Some limit Question

Let {$x_n$} be monotone increasing sequence of positive real numbers. Show that if {$x_n$} is bounded, then $\sum_{n=1}^{\infty}(1-\frac{x_n}{x_{n+1}})$ converges. On the other hand, if the sequence ...
1
vote
1answer
28 views

In Egorov's theorem, remove the condition $\mu(E) < \infty$ and let the sequence be convergent in measure. The conclusion holds for subsequence

Let $(X,\mathscr{F},\mu)$ be a measure space, $E \in \mathscr{F}$, $\{f_n\}$ is a sequence of measurable functions on $E$, and the sequence converges to function $f$ in measure. Show that $\exists ...
0
votes
2answers
51 views

Proving $\liminf_{n\to\infty}(-a_n)=-\limsup_{n\to\infty}(a_n)$

Prove: $\displaystyle\liminf_{n\to\infty}(-a_n)=-\limsup_{n\to\infty}(a_n)$ My general idea was: if $a$ is partial limit (PL) of $a_n$, then $-a$ is a PL of $-a_n$ so it follows that $s$ is the ...
1
vote
1answer
21 views

Question about calculus and real analysis

Let $x_n = \frac{p_n}{q_n} $ be asequence of rational numbers where $p_n$ and $q_n$ are coprime. Suppose also that $\lim_{n \to \infty} x_n = x $. $(x_n \neq x)$ Can we conclude that, therefore, $$ ...
2
votes
1answer
88 views

Convergence of the series of identically distributed dependent random variables

Let $a_1$, $a_2$, $\ldots$ be identically distributed, positive, not necessarily independent random variables. Consider the series $$\sum^{\infty}_{n=1} a_n$$ Is it true that the series diverges ...
0
votes
3answers
72 views

Some limit question

Suppose the series $\sum_{n=1}^{\infty} u_n$ converges. Prove that $$\lim_{n\to\infty}\frac{u_1+2u_2 +...+nu_n}{n} = 0$$ My solution is as such: $$\frac{\sum_{n=1}^{\infty} u_n}{n}\leq\frac{u_1+u_2 ...
0
votes
0answers
22 views

Gauss Test Usage

This is not really a specific math question. I am just wondering how to use Gauss Test to test if a limit of a series exists. I know the statement of the test, but I am not really sure how to use it. ...
2
votes
1answer
40 views

Is the sum of a series of continuous functions $f_n$ defined by $f(t) = \sum_{n=0}^{\infty} f_n(t)$ neccesarily continuous?

Is the sum of a series of continuous functions $f_n$ defined by $f(t) = \sum_{n=0}^{\infty} f_n(t)$ neccesarily continuous ? Let $(X,d)$ be a metric space and $f_n: X \rightarrow \mathbb C$ be ...
4
votes
0answers
77 views

Rudin's Principle of Mathematical Analysis Problem

If ${(s_n)}$ is a complex sequence, define its arithmetic means $\sigma_n$ by $$\sigma_n = \frac{s_0 +s_1 + ... + s_n}{n+1}$$ Assume that $M < \infty, |na_n| \leq M$ for all n and lim ...
1
vote
2answers
62 views

How prove this series $\sum_{n=2}^{\infty}\frac{n}{\log^{10}{n}}$ diverges?

prove or disprove this series $$\sum_{n=2}^{\infty}\dfrac{n}{\log^{10}{n}}$$ is divergent? My idea: since ...
1
vote
1answer
21 views

Let $f_n: [0,1] \rightarrow \mathbb R$ defined by $\ t \mapsto nt(1-t)^n$. Show $M_n := \sup\{ f_n(t) \mid t \in [0,1]\} = (\frac n {n+1})^{n+1}$.

Let $f_n: [0,1] \rightarrow \mathbb R$ defined by $\ t \mapsto nt(1-t)^n$. Show $M_n := \sup\{ f_n(t) \mid t \in [0,1]\} = (\frac n {n+1})^{n+1}$. I've already proved that $\{f_n\}$ is ...
0
votes
1answer
21 views

Exercise on a series

Prove the following inequality: $$\sum_{k=m+1}^{\infty} \frac{1}{k!}< \frac{1}{m*m!} \forall m\in \mathbb{N^+}$$ My strategy of attack was to set up an inequality like ...
9
votes
2answers
297 views

Does $\sum_{n=1}^\infty \frac{1}{n! \sin(n)}$ diverge or converge?

Does the series $$ \sum_{n=1}^\infty \frac 1 {n!\sin(n)}$$ converge or diverge? Even the necessary condition of the convergence is difficult to verify.
0
votes
1answer
41 views

Question on Morse inequalities

I want to understand why: if i have then $(4.1)$ is formal : it means that please help me Thank you EDIT1: $(4.1)$ tel us that $\displaystyle\sum_{q=0}^{\infty} (M_q-\beta_q)t^q=(1+t)Q(t)$ ...
4
votes
0answers
92 views

Is this Neumann series solution unique?

I have a Fredholm integral equation of the second kind given as $$f(x)=g(x)+\lambda\int_{-\infty}^\infty K(x,y)f(y)dy, $$ where $\lambda\in(0,1)$, the kernel $K(x,y)=\phi(x-y)$ is a Gaussian ...
1
vote
2answers
59 views

$\sqrt[\infty]{\infty^2}$ in limit of series using root test

I'm trying to solve a problem to show if the infinite series $\sum\limits_{k=1}^{\infty}\dfrac{k^2}{2^k}$ converges or diverges using the root test. When put in limit form, I got ...
1
vote
2answers
44 views

Infinite series: which test

I'm having troubling deciding which test to use for this: $$\sum_{n=1}^\infty (-1)^n\arctan\left(\frac{\ln(n!)}{n+4^n}\right)$$ I tried altnerating test and ratio test but I couldn't get an answer. ...
4
votes
2answers
212 views

Infinite Series with factorial

I'm having trouble manipulating the function of this series which has factorials to show that it converges or diverges using the ratio test. The series is ...
1
vote
2answers
84 views

Understanding the partition function

I have been trying, as a toy problem, to implement in either the Python or Haskell programming languages functions to calculate the partitions for a number and the count of those partitions. I have no ...
0
votes
3answers
29 views

Infinite Series Ratio test

I'm learning how to test infinite series in Calc II. I have a problem that says to use the ratio test to determine if a series converges. The series is: $\sum\limits_{k=1}^{\infty}\dfrac{k^2}{4^k}$ ...
7
votes
1answer
201 views

Series for envelope of triangle area bisectors

The lines which bisect the area of a triangle form an envelope as shown in this picture It is not difficult to show that the ratio of the area of the red deltoid to the area of the triangle is ...
7
votes
1answer
133 views

Find all natural numbers such that $\sum_{k=1}^{n} \frac{n^k}{k!}$ is an integer

Find all natural numbers such that $\sum_{k=1}^{n} \frac{n^k}{k!}$ is an integer. I've tried to bring all fractions under commmon denominator and it didn't helped me much. With guessing I find out ...
1
vote
1answer
36 views

Calculate supremum of $\left|1-e^{-\gamma_n t}\right|$

Calculate Sup (Supremum) of: $$\sup_n \left|1-e^{-\gamma_n t}\right|$$ and $$\sup_n \left(e^{-\gamma_n t}\right)$$ where $|\gamma_n|\leq M$, $M \in \mathbb R^+$ and $\gamma_n\in \mathbb R$; t is a ...
0
votes
0answers
24 views

Summing complex numbers of magnitude $1$

A professor at my university gave me the following problem: For what real values of $m$ do we have $$L=\lim_{N\to\infty}\frac{1}{N}\sum_{k=1}^Ne^{2\pi ik^3m}=0$$ If $m$ is rational, expressed ...
0
votes
2answers
31 views

Help with proof by induction?

Prove $\frac{2(n-c)}{n+1} < 2$ where c is any natural So we assume $\frac{2(n-c)}{n+1} < 2$ is true, and so far I have $\frac{2(n+1-c)}{n+2} = \frac{2n-2c+2}{n+2} = \frac{2(n-c)}{n+2} + ...
1
vote
2answers
69 views

What is that function?

What is the set of the convergence in the reals of the series $$f(x):= \sum\limits_{n=1}^{\infty} \frac 1 n \sin \left (n+\frac x n\right)?$$ Is the function $f(x)$ bounded? Edit: more exact title.
0
votes
1answer
34 views

Find a general formula for $x_k$

Suppose that the sequence $(x_k)$ is defined by $x_0 = 0, x_1 = 6, x_2 = 1$ and $$x_{k+3} = −x_{k+2}+17x_{k+1}−15x_k\quad \text{for }\, k\geq0.$$ Find a general formula for $x_k$. I have this answer ...
0
votes
2answers
41 views

Problem understanding notation

I'm learning about generating functions and in the opening explanations my book (and various sources) claim: $$a_n = 1 \forall n \in \mathbb{N}_0, \ \ \ f(x) = \frac{1}{1-x}$$. I read this as: ...
2
votes
2answers
42 views

Explore the convergence of a series

I have to explore the convergence of a series. At this picture I used radical Cauchy indication. But I don't now what to do with a denominator to find a limit. Help me please ! Thank You so much :) ...
0
votes
0answers
33 views

How to solve for absolute convergence??

Could someone help me to stepwise explain why that series doesnt converge absolutely? I used the Ratio test in order to explain the series, that is $\frac{a_{n+1}}{a_n}$ but the output doesnt makes ...
0
votes
2answers
24 views

Find a sum of a series

Help me find a sum of this series I tried to excrete as (2/7)^n * 3^(n+2) and use De Lamber indication. It gives me a result 6/7. I checked it in Wolfram Math but the result was 54. Where did I go ...
0
votes
2answers
45 views

If infinitely many points of a sequence are given, is it possible to find out the recurrence relation?

I was thinking about sequences and stumbled upon a concept. If infinitely many points of a sequence are given, is it possible to find out the recurrence relation? Please enlighten.
15
votes
1answer
232 views

Convergence of an alternating series : $ \sum_{n\geq 1} \frac{(-1)^n|\sin n|}{n}$

Study the convergence of $$\displaystyle \sum_{n\geq 1} \frac{(-1)^n|\sin n|}{n}.$$ I am stuck with this series, we need probably some measure of irrationally of $\pi$, unfortunately I am ...
0
votes
5answers
42 views

Recurrence problem for $a_5$

Assume that the sequence $\{a_0,a_1,a_2,\ldots\}$ satisfies the recurrence $a_{n+1} = a_n + 2a_{n−1}$. We know that $a_0 = 4$ and $a_2 = 13$. What is $a_5$?
2
votes
0answers
20 views

Convergence of Iteration with Sum

Consider a continuous function $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$, and the iteration $$ x_{k+1} = \frac{1}{k} \sum_{i=1}^{k} f( x_i ) $$ for some given initial condition $x_1 \in ...
1
vote
0answers
31 views

Contraction Mapping Conditions

Consider $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ continuous and uniformly bounded, and consider the iteration $$ x^{(k+1)} = \frac{1}{k} \sum_{i=1}^{k} f\left( x^{(i)} \right) + x^{(i)} $$ for ...
0
votes
2answers
28 views

Converse of absolute rearrangement theorem

If every rearrangement of a series of real numbers $\sum a_n$ is convergent , then how do we prove that $\sum |a_n|$ is convergent ?