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

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2
votes
1answer
170 views

Explaining and using the $N$-term Taylor series for $\sin x$

So I'm given the Taylor Series expansion of the sine function and I've been asked to prove it (Done) and then construct the following by my lecturer: Explain why the Taylor series containing $N$ ...
0
votes
2answers
50 views

Limit of the sequence: $\frac{k(k+1)}{(k+1)^2-qk^2}$

I am stuck on the proof of this limit: given the sequence: $$S(k,q)=\frac{k(k+1)}{(k+1)^2-qk^2}$$ $$\lim_{k\to\infty}S(k,q)=-\frac{1}{q-1}$$ How can I prove this limit?
3
votes
2answers
150 views

Proof of my conjecture on closed form of $\int _{0}^{\infty}\frac{x^{a-1}e^{-mbx}}{1-e^{-bx}}$

Let $a$, $b\in \Bbb R^+$ and $m \in \Bbb N$ then My conjectural closed form is $$\int _{0}^{\infty}\frac{x^{a-1}e^{-mbx}}{1-e^{-bx}}\,{\rm d}x = ...
0
votes
3answers
49 views

Rule for Series

I apologize if this seems really stupid, but I've been stuck in finding the general pattern for the following series: $$\sum_{n=1}^{\infty}\frac{2\cdot4\cdot6\cdots(2n)}{1\cdot3\cdot 5 ...
1
vote
2answers
221 views

Use mathematical induction to prove that for any $k \in\mathbb N , \lim (1+k/n)^n = e^k$.

Use mathematical induction to prove that for any $k \in \mathbb N, \lim (1+k/n)^n = e^k$. I already used monotone Convergence Theorem to prove $k=1$ case. Do I just need to go through the same ...
0
votes
2answers
65 views

Real Analysis: Covergence Question

Suppose that $( x_n )$ is a sequence of real numbers, $( y_n )$ is a bounded sequence of non-zero real numbers, and that $\lim x_n/ y_n = 1$. Prove that $\lim (x_n - y_n) = 0$. Since $y_n$ is bounded, ...
1
vote
1answer
320 views

How to show that a sequence is positive, monotonically decreasing and converges to 0

I have the following sequence $$ y_n = \int_0^1 \frac{x^n}{x+5}\,dx, n = 0,1,\dots $$ So first question is, how to show that it's always positive? Second is, how to show that it's ...
4
votes
0answers
123 views

Showing a sequence $x_{n+1} = Tx_n$ forms a contraction mapping

I want to show that the sequence given by $$x_{n+1} = Tx_n = x_n-\frac{(x_n^2-2)}{x_n+x_{n-1}}$$ forms a contraction mapping. That is $$|Tx_1-Tx_2|\leq c|x_1-x_2|.$$ Where $c$ is to be determined. I ...
5
votes
3answers
170 views

Does $d(x_{n+1},x_n)<d(x_n,x_{n-1})$ imply anything?

Let $(X,d)$ be a complete metric space, $(x_n)_{n\in\mathbb{N}}\subset X$ such that $d(x_{n+1},x_n)<d(x_n,x_{n-1})$ for all $n\in\mathbb{N}$. Since I cannot construct such sequence which is not ...
0
votes
1answer
64 views

Show that the sequence converges to 0 under any norm in the space (R,‖.‖) [closed]

Show that the sequence $a_n = 1/n^2$ converges to 0 under any norm in the space $(\mathbb{R},\left\| \cdot \right\|)$.
2
votes
1answer
76 views

Alternating series problem

My mind is blanking completely on how to do this one. the base string of numbers i have to pull a series out of is $$ \frac{2}{3} -\frac{2}{5} + \frac{2}{7} - \frac{2}{9} + \frac{2}{11} - ...$$ which ...
0
votes
1answer
34 views

Do greater probabilities approach expected average values with a smaller series?

Let's say you flip a coin 12 times with a goal of getting 6 heads. Then you roll a six sided die 12 times with a goal of getting 2 "ones" faces up. My intuition tells me that rolling the dice has a ...
2
votes
1answer
144 views

How find series $\sum_{n=1}^{\infty}\dfrac{(-1)^{[\sqrt[m]{n}]}}{n^a}$

let $m$ is give a positive integers, Determine for which values of $a$,the series $$\sum_{n=1}^{\infty}\dfrac{(-1)^{[\sqrt[m]{n}]}}{n^a}$$ converges where $[x]$ is the largest integer not ...
0
votes
1answer
73 views

Prove that $\sum\limits _{n=-\infty}^{n=\infty}\cos\left(2\pi nt\right)=\sum\limits _{n=-\infty}^{n=\infty}\delta\left(t-n\right)$

I've tried using Fourier transforms on both but didn't quite get anything useful. I'd really appreciate some help.
4
votes
0answers
76 views

Selecting k numbers out of N sorted numbers to a minimize a condition/Formula

A sorted list of $\mathbf N$ numbers is given. $X_1$ $\le$ $X_2$ $\le$ $X_3$ $\le$ .... $\le$ $X_N$ Select $\mathbf K$ Numbers - $Y_1$ , $Y_2$ , $Y_3$ , ..... , $Y_K$ - Such that the following ...
7
votes
2answers
165 views

What are the properties of the roots of the incomplete/finite exponential series?

Playing around with the incomplete/finite exponential series $$f_N(x) := \sum_{k=0}^N \frac{z^k}{k!} \stackrel{N\to\infty}\longrightarrow e^z$$ for some values on alpha (e.g. ...
2
votes
1answer
49 views

Series $\sum_{i=1}^\infty2^{-i}/i!$

The series $\sum_{i=1}^\infty2^{-i}/i!$ is clearly convergent by the ratio test with $\sum_{i=1}^{\infty}2^{-i}$, but is it possible to calculate the exact sum?
1
vote
1answer
17 views

The existence of inequalities between the sum of a sequence and the sum of its members

Let $(a_n)_{n=m}^\infty$ be a sequence of positive real numbers. Let $I$ denote some finite subset of $M := \{m, m+1, \cdots \}$, i.e., $I$ is the index of some points of $(a_n)_{n=m}^\infty$. Does ...
2
votes
2answers
105 views

Prove that $x_{n+1}=\frac{2}{9}(x_n^3+3)$ converges

Let $x_1=1/2$ and $x_{n+1}=\frac{2}{9}(x_n^3+3)$ for $n\geq 1$. We want to prove that the sequence $(x_n)$ converges to real number $r\in (0,1)$ satisfying the equation $2r^3-9r+6=0$. First part For ...
1
vote
0answers
95 views

Double summation of elementary functions

I am finding some trouble on calculating the following double summation: $ \sum_{k=1}^\infty \frac{b^k}{k!k}\sum_{n=0}^{k-1}\frac{(b*x)^n}{(n-1)!} $ Note that the inside sum gives: ...
3
votes
3answers
90 views

Calculating or estimating a sum of pairs of reciprocals with a constant sum

For given $M$, I would like to find $$\sum_{\stackrel{i + j = M}{i < j}} \frac{1}{i}\frac{1}{j}.$$ I'm solving the problem programatically ATM, with a single for loop for any given $M$, and I want ...
1
vote
1answer
67 views

Integral of $\ln\left(\sum_{k=0}^N a_kx^k\right)$

Given the series $$\displaystyle S(x,N)=\sum_{k=0}^N a_kx^k$$ I would like to know if exists a closed formula to calculate the undefined integral of: $$G(x,N)=\ln\left(\sum_{k=0}^N a_kx^k\right)$$ ...
13
votes
1answer
576 views

Converge or diverge? : $\sum_{n=1}^{\infty}\frac{\tan{n}}{2^{n}}$

Determine this series converge or diverge, and if it converges, find its value. $$\sum_{n=1}^{\infty}\frac{\tan{n}}{2^{n}}$$ This was too hard for me, as unboundedness of $\tan{x}$ at infinite ...
1
vote
1answer
105 views

Find a meromorphic function with given principal parts

I have the homework problem of finding an "elementary" meromorphic function $f(z)$ with the same principal parts as the sum $$\frac{1}{\pi z^2} + \frac{2}{\pi} \sum_{n=1}^\infty (-1)^n \frac{\cos ...
3
votes
2answers
83 views

Divergence of $\sum_{k=1}^{\infty}\frac{k^k}{e^k} $

I'm trying to show the series $\sum_{k=1}^{\infty}\frac{k^k}{e^k} $ is divergent by the negation of the cauchy criterion. My thought was to break the sum into dyadic pieces that could be bounded from ...
5
votes
1answer
112 views

Proving a sequence is Cauchy given some qualities about the sequence

I've got a sequence $x_n$ such that I've proved $b\leq x_n \leq c$, and $|x_{n+1}-x_{n}|\leq \frac{4}{9}|x_n-x_{n-1}|$ However I'm not very familiar with Cauchy sequences, so I don't know how to ...
5
votes
3answers
2k views

Limit of $ (2^n (n!)^2)/(2n+1)!$

I want to show that $$ \lim_{n \rightarrow \infty} \frac{2^n (n!)^2}{(2n+1)!} = 0, $$ but it's been a long time since I took calculus, and I don't know how to do it. I've tried to squeeze it, but I ...
6
votes
1answer
56 views

“Convergence” of the sequence $a_k=2^{10^{\ k}}$

I've been observing final digits of each number in the sequence $$a_k=2^{10^{\ k}}$$ You get: $\ a_0=2 \\ a_1=1024 \\ a_2= ...205376 \\a_3= ...
1
vote
0answers
52 views

Proving the convergence of a product

I have become interested in taking the $n^{th}$ term of a series and evaluating a product whose $n^{th}$ term is $(1+a_n)$. After looking around I came across the following inequality: ...
2
votes
2answers
241 views

A convergent / divergent sequence of positive numbers such that $\lim \frac{s_{n+1}}{s_n}=1$

I need to find both a convergent and divergent sequence of positive numbers such that $$\lim \frac{s_{n+1}}{s_n}=1$$ I think the question is asking me to play with the ratio test. Just when I was ...
1
vote
1answer
955 views

fourier series of $|\sin x|$

I need to find the fourier series of $$|\sin x|$$. Im not sure my way is right, would be happy if someone fix me. I found $$a_0=4/\pi$$, the function is even, so $$b_n=0$$ but how do I calculate: ...
1
vote
1answer
35 views

Mathematical series regarding complex (I think)

$\sum _{k=1}^{n-1} (n-k)\cos\frac{2k\pi}{n} $ I smell complex here...something regarding $n^{th}$ roots of unity... But I think there might be a catch...after all: ...
0
votes
1answer
26 views

Series Reduction

I'm not too familiar with sequences and series and I would like to show $ n = \dfrac{n}{2^{n-1}}(\sum\limits_{i=0}^{n-2} 2^i + 1) $ I've been playing around with it on paper and tried expanding and ...
0
votes
2answers
168 views

Explanation of this group of equalities of lim inf and lim sup

Let $(X, A, u)$ be a finite measure space ($u(X) < \infty$, e.g. $u$ is a probability measure) and $(E_n)$ a sequence of sets in A. Then $u$(lim inf $E_n$) $\le$ lim inf $u(E_n) \le$ lim sup ...
0
votes
1answer
45 views

A basic doubt on the definition of order of convergence

In the book of Luenberger, I see the following definition of order of convergence: it is defined as the supremum of the non-negative numbers $p$ satisfying $$0 \leq lim_{k -> \infty}sup ...
1
vote
0answers
70 views

lim sup of sequence of numbers vs lim sup of sequence of sets?

Let $(a_n) = (1, -1, \frac{1}{2}, 1, -1, \frac{1}{3}, 1, -1, \frac{1}{4}, ...)$ Then $\limsup (a_n) = \inf_n \sup_{k \geq n} (a_n)$ Which in this case is $1.$ However couldn't $(a_n$) be viewed as ...
0
votes
2answers
101 views

Convergence of $ \sum_{n=1}^\infty \frac{\log n}{n^q+1} $

Can anyone give a me a hint on how to check if this series is convergent or not? $$ \sum_{n=1}^\infty \frac{\log n}{n^{q}+1} $$ Thank you :)
0
votes
1answer
219 views

sum of the series $\sum e^{-n}\sin nz$

I need to Find the sum of the series $\sum e^{-n}\sin nz$ and indicate where the series converges. Make an appropriate statement about its uniform convergence. I was doing calculation like below, but ...
2
votes
1answer
60 views

Series question, telescoping.

Okay i'm starting to get a handle on this. my new question is $$\sum_{n=1}^\infty \frac{3}{n(n + 3)}.$$ I know i have to use Partial Fraction Decomposition, and what i came up with is ...
3
votes
3answers
87 views

More series help, this time a telescopic sum.

$$\sum_{n=2}^\infty \frac{2}{n^2 - 1}.$$ I Tried setting it up as a telescoping sum as $$\frac{2}{n} - \frac{2}{n-1}.$$ but now i'm sure that cannot be correct. Mayhaps I need to complete the square? ...
1
vote
2answers
36 views

Help with a Series convergence test

The problem is $$\sum_{n=1}^\infty \frac{1}{e^n} + \frac{1}{n(n+1)}.$$ I've no clue how to even being to solve this one, but the end result is $\frac{e}{e-1}$. Help would be great. This is a homework ...
0
votes
1answer
91 views

Can one use the Laurent series to find the derivatives at the center of the annulus of convergence?

I was asked to find the Laurent Series of $$f(z)=\frac{1}{z+1}-\frac{1}{z+4}$$ which I found to be, $$f(z)=\sum_{n=0}^\infty (-1)^n\frac{1}{z^{n+1}}-\sum_{n=0}^\infty (-1)^n\frac{z^n}{4^{n+1}}$$ Now ...
1
vote
3answers
36 views

I need help figuring out how this sequence converges.

$A_n = ( 1 + \frac 2n ) ^ n$; I know the end result is convergent at $e^2$, but how do I figure that out? I've started setting it up as $An = (1 + (2/n))^{(n/2)2}$ as someone has suggested but i ...
4
votes
2answers
949 views

Convergence of the arithmetic mean

Let $(a_n)_{n \in \mathbb{N}}$ be a convergent sequence with limit $a \in \mathbb{R}$. Show that the arithmetic mean given by: $$s_n:= \frac{1}{n}\sum_{i=1}^n a_i \tag{A.M.} $$ also converges to $a$. ...
2
votes
2answers
50 views

Convergence of an infinite series with doublefactorials.

$$ s(n) = \sum _{k=0}^{\infty }\left(-1 \right)^{\lfloor 1 /2\,k \rfloor } \frac{\left(n\pi \right)^{k}}{\left(2\,k \right)!! } $$ Show that $s(n)$ is an integer for all $n \ge 0$. Any idea?
1
vote
0answers
29 views

Result about Convergent Subsequences

In my lecture notes I have a proof of the following: Let $(a_{2n})$ and $(a_{2n+1})$ be subsequences of $(a_n)$. If $(a_{2n})\to l$ and $(a_{2n+1})\to l$ then $(a_n)\to l$. By definition, for ...
3
votes
1answer
124 views

Three series of Kolmogorov

Let $X_n\geqslant 0$ be a sequence of independent random variables. The following are equivalent: $i) \sum_{n=1}^{\infty}{ X_n} <\infty$ a.s $ii)$ $\sum_{n=1}^{\infty}{ \mathbb P(X_n>1)} ...
1
vote
1answer
63 views

Prove function is measurable

I was watching this exercise Let $\{a_{ij}\}\subseteq [0,+\infty]$ be a sequence. Prove that $$\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}a_{ij} = \sum_{j=0}^{\infty}\sum_{i=0}^{\infty}a_{ij}$$ If ...
0
votes
1answer
72 views

Convergent subsequence

1) Let (x_n) be a sequence and let L ∈ R. Suppose that for each ϵ > 0, {k ∈ N : x_k ∈ B(L; ϵ)} is infinite. Show that (x_n) has a subsequence converging to L. ...
1
vote
0answers
71 views

Computing a sum of an infinite series via partial decomposition

Consider the sequence defined by $u_0=0$ and $u_n=\frac{4n+1}{(2n+3)(4n^2-1)}$. The exercise asks to show that the series $\sum_{n\geq 0}{u_n}$ is convergent, which is clear since $u_n\sim_\infty ...