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

learn more… | top users | synonyms (3)

1
vote
1answer
34 views

Need some hints on a limit of sequence proof (Real Analysis)

Define $F_{0} =2 $ and $F_{n+1} =2+\frac{1}{F_{n}} $ for n=0,1,2.... Show that $F_{n+1}$ bounce back and forth on either side of its limit. Hint: assuming there is a limit, find it ...
0
votes
2answers
21 views

A question about limsup and limif

Could you please help me understand this question: Suppose $a_n$ is bounded sequence and $A<\liminf a_n$, $B>\limsup a_n$. Prove : $A<a_n<B$ for all n>N. It seems to me to simple to be ...
0
votes
2answers
21 views

Calculate the following sequence $\sum_{n=0}^{+\infty }\left ( -\dfrac{1}{4\alpha } \right )^{n}\dfrac{ (2n)!}{n!},\; \alpha >0$

Calculate the following sequence $$\sum_{n=0}^{+\infty }\left ( -\dfrac{1}{4\alpha } \right )^{n}\dfrac{ (2n)!}{n!},\; \alpha >0$$
9
votes
3answers
567 views

Is this already an equation/law that has been found?

So I was messing around with some numbers today and I have found a way to quickly add summations (probably not the first one to discover it but...) this only works when you start at 1 (i.e. ...
1
vote
1answer
78 views

Evaluate $\sum\limits_{n\mathop=1}^{\infty}\frac{\sin2\pi nx}{\pi n}$

Prove $$x-\lfloor x\rfloor-\frac{1}{2}=-\sum_{n\mathop =1}^{\infty}\frac{\sin(2\pi nx)}{\pi n}$$ where $x$ is any non-integer real number.
1
vote
1answer
38 views

Real Analysis question on sequences (Hint needed!!!)

Given the number $\alpha > 1$ , define the sequence an where $a_0 = 1$ and $a_{n+1} = (\alpha \times a_{n})^{\frac{1}{4}}$ for $ n \geq 0 $. Prove: If $a_{n}^{3}< \alpha $(as is true when n ...
1
vote
1answer
30 views

Prove that the following sequence converges and find its limit

Prove that the following sequence converges and find its limit $$\frac{1}{3+\frac{1}{3}},\frac{1}{3+\frac{1}{3+\frac{1}{3}}},\frac{1}{3+\frac{1}{3+\frac{1}{3+\frac{1}{3}}}},\ldots$$ I started to ...
0
votes
0answers
7 views

normality of set of analytic functions whose derivative is normal

I have this question from old preliminary exam problem set. (a) Show that if F⊂H(G) is normal then F′={f′:f∈F} is also normal. (b) Does F⊂H(G) normal imply F′={f: f′∈F} is normal? Otherwise give a ...
2
votes
1answer
30 views

When are these series equal?

Suppose we have a power series $$\sum_{n=0}^\infty {a_nb_nx^n}$$ When is it true that the series obtained by eliminating $b_n$ is proportional to the original series? $$\sum_{n=0}^\infty ...
2
votes
2answers
71 views

If $a_{n+1}=1+\frac{1}{a_{1}+a_{2}+\cdots+a_{n}-1}$ then $0<a_{n}<1$

define the sequence $\{a_{n}\}$ and such $$a_{1}=a,a_{n+1}=1+\dfrac{1}{a_{1}+a_{2}+\cdots+a_{n}-1},n\ge 1$$ Find the all real number $a>0$,such $$0<a_{n}<1,n\ge 2$$ My try: since ...
14
votes
2answers
173 views

Conjecture: the sequence of sums of all consecutive primes contains an infinite number of primes

Starting from 2, the sequence of sums of all consecutive primes is: $$\begin{array}{lcl}2 &=& 2\\ 2+3 &=& 5 \\ 2+3+5 &=& 10 \\ 2+3+5+7 &=& 17 \\ ...
0
votes
2answers
22 views

Proof for a series given

Prove that $$\frac{z}{(1-z)^2} = \sum_{n=1}^{\infty} nz^n.$$ Do I need to do this by induction or by any other way? Please help.
0
votes
1answer
12 views

$ (1-x)^2\sum_{N+1}^\infty (n+1)x^n = x^{n+1}(n+2 - (n+1)x) $ small independent of $x\in (0,1)$?

In this post a user made the following claim: Claim: Suppose $\rho_n$ is a sequence of real non-negative numbers converging to $0$. Suppose $x\in (0,1)$. Then $$ \sum_{N+1}^\infty ...
-1
votes
0answers
12 views

On sequence and function [on hold]

For a set $S$ we denote its cardinality by $|S|$. Let $e_1,e_2,\ldots,e_k$ be non negative integers. Let $A_k$ (respectively $B_k$) be the sets of all $k$-tuples $(f_1,f_2,\ldots,f_k)$ of inteers such ...
0
votes
1answer
50 views

Suppose that $\sum a_i$ converges and that $a_i\geq0$ for all 𝑖.

Suppose that $\sum a_n$ converges and that $a_n\geq0$ for all $n$. For each $n$, let $e_n=\pm1$. Then, prove that $\sum e_na_n$ converges. Can I simply say that ∑|eᵢaᵢ| = ∑aᵢ so that ∑eᵢaᵢ converges ...
-2
votes
1answer
28 views
2
votes
1answer
31 views

asymptotics of this sum $ x \to 0 $

given the sum $$ \sum_{n=0}^\infty \frac{\exp(-nx)}{n+a} =f(x) $$ what would be the asymtptic of this series ?? for $a=1$ i believe this series goes as $ f(x) \sim \frac{1}{x}+ \gamma $ for every ...
-2
votes
1answer
48 views

$\sum r 2^r$ series computation [on hold]

How do you compute this series? $$ \sum_n n 2^n $$ I tried it the same way as computing the geometric series but I didn't come to a result Hope someone could help Edit The series goes from 0 to ...
0
votes
2answers
43 views

summation of ceil and floor function

I need a closed solution or a faster algorithm for calculating $$ \sum_{k=1}^{n-1} \left\lceil \frac{n}{k}-1 \right\rceil $$ and $$ \sum_{k=1}^{n-1} \left\lfloor \frac{n}{k} \right\rfloor $$ where $ ...
2
votes
5answers
269 views

Finding generating function for the recurrence $a_0 = 1$, $a_n = {n \choose 2} + 3a_{n - 1}$

I am trying to find generating function for the recurrence: $a_0 = 1$, $a_n = {n \choose 2} + 3a_{n - 1}$ for every $n \ge 1$. It looks like this: $a_0 = 1$ $a_1 = {1 \choose 2} + 3$ $a_2 = {2 ...
2
votes
1answer
34 views

${2+6\over 4^{100}}+{2+2\cdot6\over 4^{99}}+{2+3\cdot6\over 4^{98}}+\cdots+{2+99\cdot6\over 4^2}+{2+100 \cdot6\over 4}$

Find the value of $${2+6\over 4^{100}}+{2+2\cdot6\over 4^{99}}+{2+3\cdot6\over 4^{98}}+\cdots+{2+99\cdot6\over 4^2}+{2+100\cdot6\over 4}$$ My approach: $${2\over 4^{100}}+{2\over ...
2
votes
2answers
53 views

find sum of first 2002 terms

if $\left \{ a_n \right \}$ is sequence of Real Numbers for $n \ge 1$ such that \begin{equation} a_{n+2}=a_{n+1}-a_n \tag{1} \end{equation} \begin{equation} \sum_{n=1}^{999} a_n=1003 \tag{2} ...
0
votes
1answer
28 views

Compute the following sum $S_n(x,a):=\displaystyle\sum\limits_{k=0}^{n}\binom{n}{k}a^k\cos(kx)$

Compute the following sum $S_n(x,a):=\displaystyle\sum\limits_{k=0}^{n}\binom{n}{k}a^k\cos(kx)$ From the binomial theorem $(1+z)^n=\sum_{k=0}^n\binom{n}{k}z^k$ Hence ...
1
vote
2answers
40 views

Prove the series $\sum_{n = 1}^{ \infty} \frac 1 6 n (\frac 5 6)^{n-1} = 6$

Prove the series $$\sum_{n = 1}^{ \infty} \frac 1 6 n (\frac 5 6)^{n-1} = 6$$. I've tried various methods for proving the series: The series is not geometric, but I see that $\frac 1 6 n (\frac ...
1
vote
3answers
50 views

Finding the limit of a sequence by diagonalising a matrix

Consider the sequence described by: $\frac11 , \frac32 , \frac75 , ... ,\frac {a_{n}}{b_{n}}$ where $ a_{n+1} = a_n +2b_n $ and $b_{n+1} = a_n+b_n$ Find a matrix $A$ such that ...
0
votes
2answers
27 views

Determining the convergence of a series

The series I have is: $$\sum^{\infty}_{n=1} (-1)^{n} \int^{\pi}_{0} (x+n\pi)^{-p}\sin{(x)} dx$$ where $p>0$. We can use the alternating series test where $$a_{n} = \int^{\pi}_{0} ...
4
votes
2answers
114 views

Doubly infinite sequence limit

I have a 2-indexed sequence $F^n_m$ where $n,m$ are natural numbers and I am concerned about the behavior as $n\to\infty$ and/or $m\to\infty$. The sequence is expressed as ...
5
votes
3answers
386 views

Is there a formal definition of convergence of series?

One is often asked to check if a given series converges or not in one's first year of uni. Is there a formal definition that allows us to check this? We are only given a bunch of tests that are ...
2
votes
1answer
45 views

Finding another series for a given series.

For each sequence $$a_n$$ find a number r such that $$\frac{a_n}{r^n}$$ has a finite non-zero limit. (This is of use, because by the limit comparison test the ...
-4
votes
0answers
39 views

Which of the following is correct? [on hold]

Let $$X = \frac{1}{1001} + \frac{1}{1002} + \frac{1}{1003} + \cdots + \frac{1}{3001}$$ Then find the correct option. (A) $X < 1$ (B) $X > \frac{3}{2}$ (C) $1 < X < \frac{3}{2}$ (D) ...
0
votes
0answers
20 views

Find if this sequence converges or diverges?

I'm trying to determine whether or not this sequence converges or diverges: However, I'm quite unsure as to where to begin. I believe that it diverges if we take the limit, however, is this the way ...
0
votes
0answers
16 views

Does this recursion/sequence of iterated infinite sums converge?

Let $n,x\in\mathbb{N}$, $\alpha,\beta,\lambda\in\mathbb{R}^+$, where $\alpha,\beta<1$. Does the following sequence converge (and to what)? $s_0=\alpha n+\lambda$ ...
2
votes
1answer
72 views

Convergence of “alternating” harmonic series where sign is +, --, +++, ----, etc.

Exercise 11 from section 9.3 of Introduction to Real Analysis (Bartle): Can Dirichlet’s Test be applied to establish the convergence of $$ 1 - \dfrac12 - \dfrac13 + \dfrac14 + \dfrac15 + ...
0
votes
1answer
46 views

Series $\sum_{k=1}^\infty \frac{k^k}{e^k(k+1)!}$

Does the series $$\sum_{k=1}^\infty \frac{k^k}{e^k(k+1)!}$$ converge? Mathematica says it converges by comparison test. But I don't see how.
0
votes
2answers
28 views

Determining Divergence

How can I prove that this series diverges? I don't think you can use a comparison test, but maybe I'm mistaken. $$\sum \dfrac 1{n^{4/5}+10^{10}}$$
3
votes
3answers
109 views

determine if $\sum^{\infty}_{n=1} \frac{n}{e^n}$ converges

determine if $\sum^{\infty}_{n=1} \frac{n}{e^n}$ converges. I used the ratio test and eventually came to that: $\frac{n+1}{en}$ which is approximatley equal to $\frac{1}{e}$ which is less than 1. ...
0
votes
1answer
18 views

Taylor Polynomials

My question is this, Compute $T_2(x)\ $ at $x=0.8$ for $y=e^x$ I have figured out at that $T_2(x)$ equals: $$e^{0.8}+e^{0.8}(x-0.8)+\frac{e^{0.8}((x-0.8)^2)}{2}$$ The second part of the question ...
2
votes
3answers
29 views

Help with sequence problem, expressing it as a function of a?

I'm working on a problem set for a math course right now and I've come across a problem that I am having some difficulty understanding/solving. The problem is below: Consider the sequence: for ...
0
votes
2answers
72 views

How to evalute this limit?

Let $A_N$ defined as follow: $$A_N=\prod_{n=1}^{N}\dfrac{\alpha n+\nu}{\beta n+\mu}.$$ Calculate: $$\ell=\lim_{N\to\infty}A_N.$$ P.S. This not a homework. I just thought about it and I tried to ...
1
vote
1answer
29 views

prove $\lim_{n\to\infty}\sup\{|f(x) - f_n(x) | : x \in S \} =0$

I understand how the theorem works but how would you prove that a sequence $f_n$ of functions on set $S \subset \mathbb{R}$ converges uniformly iff $$\lim_{n\to\infty} \sup\{|f(x) - f_n(x) | : x \in ...
0
votes
1answer
29 views

German and French word for Basic sequence

Wikipedia says that a sequence $(x_n)$ is a basic sequence iff it is a Schauder basis of its closed linear span. I was wondering whether there is a French or German word for 'basic sequence'? How ...
0
votes
2answers
1k views

Prove: Convergent sequences are bounded

I don't understand this one part in the proof for convergent sequences are bounded. Proof: Let $s_n$ be a convergent sequence, and let $\lim s_n = s$. Then taking $\epsilon = 1$ we have: $n > N ...
0
votes
0answers
7 views

Particular type of input to gather information on certain types of encryption schemes

Consider a simple case in which information is sent is considered as it's binary equivalent and then those numbers are considered as base 10 and used as inputs to an equation over a number field of ...
2
votes
1answer
38 views

Sum of fractions of squared sines

I'm trying to prove the following approximate identity for $p$ integer: $$ \sum_{l=1}^m\frac{\sin^2\left(\frac{\pi l}{p}\right)}{\sin^2\left(\frac{\pi l}{mp}\right)}\sim \frac{m^2(p-1)}{2}+O(m) $$ ...
1
vote
1answer
33 views

Sequence bounds and limit

I'm doing the following exercise. Given the sequence \begin{cases} a_{n+1} = {n + 8\over4n + 1}*a_n & n=0, 1, 2 \\ a_{0} = 1 \end{cases} Find if the sequence is definitely decreasing/increasing. ...
0
votes
1answer
18 views

Fixed Point Iterations for Bounded Affine Functions

Let $f: X \rightarrow X$ be continuous and $X \subset \mathbb{R}^n$ compact and convex. From Brouwer fixed-point theorem, $f$ admits a fixed point ($\bar{x} \in X$ such that $f(\bar{x}) = \bar{x}$). ...
0
votes
3answers
54 views

Limit of the summand

I learnt that if $\displaystyle\lim\limits_{x\mathop\to\infty}f(x) \ne 0$ or if the limit does not exist then $\displaystyle\sum_{x\mathop=1}^{\infty}f(x)$ diverges. But suppose $f(x)$ takes the ...
0
votes
1answer
11 views

Multiplying non-decreasing sequences

Let $(a_n)$ and $(b_n)$ be non-decreasing sequences of positive terms (i.e. $a_n\gt0$ and $b_n\gt0$ for all $n\ge1$). Prove that the sequence $(c_n)$ is non-decreasing, where $c_n=a_nb_n$ for all ...
0
votes
3answers
31 views

Convergance of a sequence [on hold]

Prove that the sequence $(a_n)$ converges, where$$a_n=\frac {3+n+4{n^2}}{1-n+3{n^2}}$$ for all $n\ge1$
0
votes
1answer
276 views

Prove that absolute convergence implies unconditional convergence

In the proof of "absolute convergence implies unconditional convergence" for a convergent series $\sum_{n=1}^{\infty}a_n$, we take a partial sum of first $n$ terms of both the original series ($S_n$) ...