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

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0
votes
2answers
24 views

Increasing/ Decreasing Annuity

Can we evaluate $c$ such that the following series converges using the definition of limit of a sequence? $$ s{_n}=\sum_{t=1}^{n}tc^{t}, c\in \mathbb{R} $$
0
votes
2answers
51 views

Solve Fibonacci-like linear recurrence equation

How to solve the following equation: $f(n) = f(n-1) + f(n-2) + 1$ My best guess is that it has something to do with Linear Recurrence Equation. I know how to do it without the constant $1$, which ...
1
vote
1answer
84 views

Intuition behind a convergent subsequence of $\sin(n)$

$\let\eps\varepsilon$ I was looking through a Real Analysis exam paper one day and was stuck on a question; fortunately there is a solution provided which I will sketch below, but I have no intuition ...
2
votes
1answer
66 views

Infinite series containing binomial coefficients

I've encountered the following series: $$\sum_{t=1}^\infty {1 \over 2^{t}}\, {{\large t} \choose {\large{t + x \over 2}}}$$ Is this series even convergent? I'm really lacking knowledge on series ...
0
votes
4answers
82 views

Does the series diverge or converge?

Does the series $\sum \frac{1}{n\sqrt{n^2+1}}$ diverge or converge? When I try to solve it I use the limit comparison test. It all works well when I compare it to the function $\frac{1}{n^2}$. The ...
4
votes
2answers
169 views

Does $\sin(\sin(\sin\cdots(\sin1)\cdots) \rightarrow 0 $?

Stuck on homework problem (not this), if I can prove as a lemma that the sequence $$\sin(\sin(\sin\cdots(\sin1)\cdots) \rightarrow 0 $$ then I'm done. It's monotonic and decreasing and bounded by 0 ...
11
votes
1answer
339 views

Forcing series convergence

I am trying to figure this out: $\mathscr{S}=\big\{(a_n),(b_n),\dots \big\}$ is a finite set of real, null sequences. Does there exist a sequence $(\epsilon_n)$, where $\epsilon_k=\pm 1$ for each ...
1
vote
1answer
30 views

Problems with fixed point equotation regarding the sequence $\left( \sqrt[n]{n} \right)$

Let $a_n=\sqrt[n]{n}$ then we find $a_n=( (a_{n-1})^{n-1}+1)^{\frac{1}{n}}$. Therefore we can make a fixed point equotation to find the limit $a$ of the sequence $(a_n)_{n\in \mathbb{N}}$. $a = ...
0
votes
1answer
48 views

Sequence of integers

The sequence of integers $\left\{a_{n}\right\}$ is defined by $$ a_{0}= 1\,,\quad a_{1}=2\,,\qquad a_{n + 2}= a_{n} + a_{n+1}^{2}\quad\mbox{for}\quad n \geq 0 $$ What is the rest in the euclidean ...
2
votes
0answers
40 views

Show that uniform continuity implies stochastic equicontinuity

Let $\Theta$ be a metric space and assume it is compact. Let $W_t: \Omega \rightarrow \mathbb{R}^k$ be a random variable for $t\leq T$. Let $m(.,\theta): \mathbb{R}^k\rightarrow\mathbb{R}^s$. Let ...
3
votes
1answer
121 views

What is the sum of $\sum\limits_{n = 1}^\infty {\arctan \dfrac{2}{{{n^2}}}}$ and why? [duplicate]

The series $$\sum\limits_{n = 1}^\infty {\arctan \dfrac{2}{{{n^2}}}}$$ converges because it is asymptotic to $\dfrac{2}{n^2}$ which is convergent. What is its sum and why?
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votes
1answer
42 views

Express the coefficients of $\exp(f(x))$ using the coefficients of $f(x)$

Given a formal series $$f(x)=\sum_{k=1}^\infty f_k x^k$$ what is $$K_n:=\left[\left(\frac{d}{dx}\right)^n e^{f(x)}\right]_{x=0}$$ in terms of the coefficients $\{f_k\}$? I stumbled upon this ...
0
votes
1answer
60 views

Binomial coefficient - first two terms, proof of inequality

I've seen the following and I'm not sure whether it is true or not, and if yes, why it holds. $(1-p)^x \geq 1-p x$ for $p\in (0,1)$ and $x>0$. Do I need some additional Information to prove ...
0
votes
1answer
29 views

Derive a formula to get the particular value from table

I have a table of points earned given the final game score. ...
2
votes
1answer
217 views

if $a_n=\frac{a_{[\frac{n}{2}]}}{2}+\frac{a_{[\frac{n}{3}]}}{3}+\ldots+\frac{a_{[\frac{n}{n}]}}{n}$,then $a_{2n}<2a_{n}$

Question: Consider the following sequence : $$a_1=1 ; a_n=\frac{a_{[\frac{n}{2}]}}{2}+\frac{a_{[\frac{n}{3}]}}{3}+\ldots+\frac{a_{[\frac{n}{n}]}}{n}$$. Prove that: $$a_{2n}< 2a_{n } (\forall ...
1
vote
1answer
33 views

Proving this series is convergent almost everywhere

Let $\{a_n\},\{r_n\}$ be two sequences of real numbers such that $\sum_{n=1}^\infty|a_n|<\infty$. Prove that $$\sum_{n=1}^\infty\frac{a_n}{\sqrt{|x-r_n|}}$$ converges absolutely almost everywhere ...
0
votes
2answers
30 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
44 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
134 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
28 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
48 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
43 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
381 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 ...
2
votes
1answer
54 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
79 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
74 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
32 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.
12
votes
5answers
644 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 ...
1
vote
1answer
91 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
55 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 ...
2
votes
2answers
84 views

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

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
25 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
252 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
80 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 ...
2
votes
1answer
69 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
105 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
66 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
23 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
22 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 ...
10
votes
2answers
326 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
43 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 $\displaystyle\sum_{q=0}^{\infty} (M_q-\beta_q)t^q=(1+t)Q(t)$ ...
5
votes
0answers
202 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
69 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
48 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. ...
5
votes
2answers
663 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 ...
2
votes
2answers
136 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
38 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: $\sum\limits_{k=1}^{\infty}\dfrac{k^2}{4^k}$ ...
8
votes
1answer
261 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 ...
8
votes
1answer
143 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
38 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 ...