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

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0
votes
5answers
44 views

Sum $S=a_1+a_2+…+a_{2015}$

Given positive real numbers $a_1,a_2,...,a_{2015}$ whose product $\prod_{i=1}^{2015} a_i=1$. What can you say about their sum $S=a_1+a_2+...+a_{2015}$ $S $ can be any positive number $1\leq S \leq ...
2
votes
2answers
59 views

$\sum \frac {1}{n^2 a_n}$ is divergent

Suppose $a_n\geq 0$ and $\sum a_n$ converges .Show that $\sum \frac {1}{n^2 a_n}$ is divergent I think that this inequality will going to help me. $\frac{1}{n^2a_n} \leq \frac{1}{n(na_n)} \leq ...
0
votes
0answers
26 views

Simplify of linear sequence [duplicate]

I have the following sequence: $$n + (n-1) + (n-2) + (n-3) ... + 3 + 2 + 1 + 0$$ I know by definition that the above sequence can be simplify to: $$\frac{1}{2} n(n+1)$$ What I can't understand is ...
5
votes
1answer
55 views

converge value of series $\sum_{n=0}^{\infty} \left( \frac{1}{n+d+1} - \frac{1}{n+5d+1} \right) $

\begin{align} \sum_{n=0}^{\infty} \left( \frac{1}{n+d+1} - \frac{1}{n+5d+1} \right) = \sum_{n=0}^{\infty}\frac{4d}{(n+d+1)(n+5d+1)}= ? \end{align} I know from the $p$-test, ($i.e$ $\sum \frac{1}{n^p}$ ...
1
vote
0answers
23 views

compact convergence for a series in complex space

I need some help with this. I have to show that the follwing series converges compat. $$\sum_{n=1}^\infty f_n :D:= \{z \in \mathbb{C} | Re(z) > 0 \} \to \mathbb{C}, f_n (z):=\frac{1}{z+n^2} $$ I ...
0
votes
2answers
28 views

Linear Recurrences

I was working on linear recurrence... But I am having trouble with it. $a_0 = 1; a_1 = 2; a_k = 4a_{k-1} - 2a_{k-2}$ I found $a_2$ which is $$4a_1 - 2 a_0=4\cdot2 - 2\cdot 1 = 6$$ I found $a_3$ ...
-1
votes
0answers
45 views

Use the Lipschitz estimate to prove [closed]

Let ($f_n$) be a sequence of functions that are continuous on $[a, b]$ and differentiable on $(a, b)$. How to Use the Lipschitz estimate to prove that $|f_n(x) - f_p(x) - (f_n(c) - f_p(c))| \leq ...
4
votes
3answers
113 views

An infinite series

Hopefully this question is not a duplication. Consider the following infinite series: $$\LARGE\sum_{k=0}^\infty\frac{2^k}{1+\frac{1}{x^{2^k}}}$$ We know the answer is $\frac{x}{1-x}$ if ...
0
votes
2answers
44 views

infinite series and proof of sum using induction.

Consider the series: $$ \sum_{i=1}^\infty \frac{i}{(i+1)!} $$ Make a guess for the value of the $n$-th partial sum and use induction to prove that your guess is correct. I understand the ...
0
votes
2answers
65 views

l'Hopitals Rule for limit

To use the l'Hopitals rule to evaluate the limit of $\lim_{x \rightarrow 1} \dfrac{1 - x + \ln x}{1 + \cos(\pi x)}$ I get an undefined answer! My working is: If we let: $f(x) = 1 - x + \ln x$, ...
0
votes
2answers
59 views

A series with the recursive formula.

A sequence $\lbrace a_{n}\rbrace_{n\geq 0}$ is constructed by choosing a value of $a_{n}$, and then the following elements are determined from the equation $a_{n}=2-\frac{1}{2}a_{n-1}$ for ...
0
votes
2answers
35 views

Expansion of reciprocal of quadratic

Can I expand $\frac{1}{1-.7B-.3B^2}$ into an infinite series? Where B is the backwards operator in time series. I was thinking $\frac{1}{1-(-.3B)}\frac{1}{1-B}$. Express this as a product of a ...
1
vote
1answer
40 views

A contest problem from a Regional math tournament

For an integer $n >2$ show that $$\frac{1}{2}\sqrt{1+\frac{1}{n}} < \sum_{k=1}^{n}\frac{1}{\sqrt{n^2+k}}$$. I tried to integrate the RHS and got $$2n\left(\sqrt{1+\frac{1}{n}}-1\right)$$ But do ...
1
vote
1answer
38 views

Need a hint for convergence of alternating series

So I have this following sum $1-\frac{\pi^2}{2!} + \frac{\pi^4}{4!} - \frac{\pi^6}{6!} + \space ... \space $ Now obviously this is equal to the following infinite sum $$\sum_{n=0}^\infty (-1)^n ...
0
votes
2answers
67 views

How can I prove that $\frac{(3^n + 4^n)}{(4^n + 1)}, n \in \mathbb{N}$ is bounded?

How can I prove that $$\frac{(3^n + 4^n)}{(4^n + 1)}$$ $n \in \mathbb{N}$ is bounded? Please don't tell me the full answer, I would just like a push in the right direction.
9
votes
6answers
309 views

Definition of a geometric sequence

Is the sequence $0, 0, 0, 0 ...$ geometric? If so how would you define it? In order to define a geometric sequence you need the first term, and the ratio of terms. In this case you could have: $a = ...
0
votes
0answers
46 views

$\sum a_n$ converges $\implies\ \sum \sqrt{a_na_{n+1}}$ converges?

Let $a_n > 0.$ When $\sum a_n$ converges $\sum \sqrt{a_n a_{n+1}}$ converges or not? For, $$\frac{\sqrt{a_n a_{n+1}}}{a_n}=\frac {\sqrt{a_{n+1}}} {\sqrt{a_n}}$$ $\because$ By comparison test ...
0
votes
1answer
20 views

Does $\lVert\mathbf{x}^{(n)}-\mathbf{x}^{(n-1)}\rVert_2\rightarrow0$ imply convergence of $\mathbf{x}^{(n)}$?

A sequence $\{\mathbf{x}^{(n)},n=1,2,...\}$. If $\lVert\mathbf{x}^{(n)}-\mathbf{x}^{(n-1)}\rVert_2\rightarrow0$, does it also imply the convergence of the whole sequence $\{\mathbf{x}^{(n)}\}$?
0
votes
0answers
46 views

Countability of $\mathbb{Q}$ using two sequences. [closed]

I'm trying to prove the countability of $\mathbb{Q}$ using the following problem. For integers $n\ge 1$ , let $a_n=k$ if $3^k$ divides $n$ and $3^{k+1}$ doesn't divide $n$. Let $b_1=2$, and for ...
2
votes
2answers
60 views

How to calculate the Summation??

Can we get the formula in terms of N and k for this summation series? $$ A=\sum_{t=0}^N\sum_{s=0}^t\sum_{r=0}^sk^rk^{s-r}k^{t-s} $$
0
votes
0answers
24 views

How can I represent a fractal fraction in a way that can control precision?

I'm looking for shorter ways to represent a fractal fraction where the value can be found at a level of precision ($p_n$), similar to the following example, but without expanding the whole fraction: ...
0
votes
2answers
51 views

Prove the convergence of series $\sum_{k=1}^{\infty}\log(1+\frac{1}{\sqrt{k}})$ by Cauchy criterion

Given $$\sum_{k=1}^{\infty}\log\left(1+\frac{1}{\sqrt{k}}\right)$$ and by definition I need to prove that for $\forall \epsilon>0, \exists n_0 \text{ s.t. } \forall n>n_0, \forall p=1,2,...$ ...
1
vote
1answer
20 views

Find a formula for $\langle X_n\rangle$ which is defined recursively as follows

$X_1=a$, $X_2=b$ and $X_{n+2}=(X_n+X_{n+1})/2$ Find a formula for $\langle X_n\rangle$ valid for each $n\in\mathbb N$. I wrote a few terms in this sequence and tried to derive a formula. But I ...
2
votes
1answer
58 views

The series of function $f(x)=\sum_{n\geq 1}\frac{1}{n}\ln(1+\frac{x}{n})$; the convergence and the differentiability.

Consider the series of function $f(x)=\sum_{n\geq 1}\frac{1}{n}\ln(1+\frac{x}{n})$ for $x>-1$. a) Show that the series is pointwise convergent. Answer: I actually don't know how to show ...
2
votes
1answer
19 views

Uniform convergence of $(f_n(x)) = \frac{nx}{n+x}$ on $I=[0,1]$

Test whether or not $(f_n(x)) = \frac{nx}{n+x}$ on $I=[0,1]$ converges uniformly on $I=[0,1]$. My attempt: $\displaystyle \lim_{n \to \infty}f_n(x) = \lim_{n \to \infty}\frac{nx}{n+x} = \lim_{n ...
2
votes
7answers
117 views

$\sum a_n$ converges $\implies\ \sum a_n^2$ converges? [duplicate]

If $\sum a_n$ with $a_n>0$ is convergent, then is $\sum {a_n}^2$ always convergent? Either prove it or give a counter example. Im trying in this way, Suppose $a_n \in [0,1] \ \forall\ n.\ $ Then ...
2
votes
4answers
205 views

Showing a particular recurrence is constant

A sequence, $ ( a_n ) _ { n \in \mathbb{N}} $, is constructed by selecting a value of $ a_0$, and then successively forming the following elements from the equation. $$ a_n = 2- \frac12 a_ { n- 1} ...
4
votes
1answer
23 views

Does $(f_n(x))= (\frac{nx}{1+nx^2})$ converge pointwise/uniformly on $I= [0,1]$?

Does $\displaystyle(f_n(x))= \bigg(\frac{nx}{1+nx^2}\bigg)$ converge pointwise/uniformly on $I= [0,1]$? My attempt: Pointwise: $\displaystyle \lim_{n \to \infty}f_n(x) = \lim_{n \to \infty} ...
0
votes
1answer
13 views

on the convergence of an infinite series involving logarithms

It looks like the following quantity $$ q(k)=\frac{k+1}{2k}(1+\log k) - \sum_{i=2}^k \frac{i}{k^2} \log i $$ tends to $3/4$ as $k$ goes to infinity. Is there a nice way to prove it?
1
vote
2answers
24 views

Evaluation of Infinite Series

Using the fact that $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} = \log 2$, then evaluate $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n(n+1)}$. I tried using partial fractions by splitting $\frac{1}{n(n+1)}$ ...
1
vote
1answer
30 views

How to find a formula for these generating sequences?

It is given that $a_{0}=1$ , $b_{0}=0$ , $c_0=0$ $$ c_n= xc_{n-1}+x(x-1)a_{n-1}(3b_{n-1}+(x-2)a_{n-1}^{2})) $$ $$ b_n=xb_{n-1}+x(x-1)a_{n-1}^{2} $$ $$ a_n=xa_{n-1}+1 $$ where x is any constant. ...
1
vote
0answers
27 views

Finding an explicit expression for a sequence

I am looking for a method (if it even exists) to find the closed form for sequences like : $u_{n+2}=\sqrt{\alpha u_{n+1}+ \beta u{n}}$ with $\alpha,\beta,u_0,u_1>0$ I already know a method exists ...
47
votes
4answers
2k views

Why is a geometric progression called so?

Just curious about why geometric progression is called so. Is it related to geometry?
1
vote
1answer
11 views

Simple question: uniform convergence with constant terms

If I have a sequence that is made up of two part: one that varies with $n$ and another that remains constant, does uniform convergence of the part that varies imply uniform convergence of the whole ...
1
vote
1answer
45 views

limit of jacobi theta 2 or simple series

I have a simple problem: I need to evaluate the limit $x\rightarrow 1$ of the Jacobi Theta function 2 $$\Theta_2(m,x)=2x^{1/4}\sum_{k=0}^\infty x^{k(k+1)}\cos((2k+1)m)$$ when $m=0$, that to say ...
6
votes
3answers
105 views

$\sum_1^{\infty} \frac{(p+1)(p+2)(p+3)…(p+n)}{(q+1)(q+2)(q+3)…(q+n)}$ convergence

I need to determine for which values of $p$ and $q$, both greater than $0$, the following series converges: $$\sum_1^{\infty} \frac{(p+1)(p+2)(p+3)...(p+n)}{(q+1)(q+2)(q+3)...(q+n)}$$ I've tried ...
0
votes
5answers
49 views

How can I solve the progression?

Sum of first three member of arithmetic progression equal $21$. If from one and two first members deduct $1$ and third member add $2$, it will be a geometric progression. first member $$a_1 - ...
1
vote
0answers
35 views

Elementary proof about nth differences of nth powers of integer

In a post on Math.SE., a proof sketch was proposed for the proposition below: The sequence of $n$th differences of the sequence of $n$th powers of positive integers, is the constat sequence $n!$. ...
5
votes
2answers
176 views

Finite Summation of Fractional Factorial Series

Is there a closed form solution for the following series? (Without Using Gamma Function): $$ S=\sum _{i=1}^{n-1} \frac{1}{(i+1)!} $$
0
votes
2answers
24 views

How does one arrive at a certain expression for the Fibonacci Zeta function?

In this paper by L. Navas, it is described how one can obtain a analytic continuation of the Fibonacci Dirichlet series (though I'm not sure it's actually a Dirichlet Series). First, the following ...
-1
votes
1answer
42 views

Two hard series problem [on hold]

I have to calculate $\sum_{n=0}^{\infty}\frac{n}{2^n}$ and $\sum_{n=0}^{\infty}\frac{n^2}{7^n}$. I tried to rewrite is as $-\ln(1-x)=\sum_{r=1}^\infty\dfrac{x^r}r$ but I only get this ...
2
votes
0answers
26 views

Division of two series expansions

I have the two functions $u(x)$ and $v(x)$, both of which have known basis expansions $u(x) = \sum_n a_n f_n(x)$, $v(x) = \sum_n b_n f_n(x)$. I would like to calculate the function ...
0
votes
1answer
45 views

Help with discrete mathematics proof

I am to prove $A_0\cap(\bigcup_{i=1}^n A_i) = \bigcup_{i=1}^n (A_0\cap A_i), n\ge 2$ by induction. I started out like this: Step 1: Prove that $A_0\cap(\bigcup_{i=1}^n A_i) = \bigcup_{i=1}^n ...
3
votes
2answers
30 views

Limit of sequence and Riemann sum

I have to calculate $$\lim_{n \to \infty}\sum_{k=1}^{n}\frac{(k-1)^7}{n^8}$$ So, $$\lim_{n \to \infty}\sum_{k=1}^{n}\frac{(k-1)^7}{n^8} = \lim_{n \to ...
7
votes
1answer
83 views

Show that $ \sum_{n\in \mathbb {S}} \frac{1}{n} $ is convergent [duplicate]

Let $\mathbb {S} =\left \{ 1,2,3,...,9,11,12,...,19,21,...99,111,112,113... \right \} $ i.e, the positive integers set which contain zero digit is omitted. Now show that $ \sum_{n\in \mathbb {S}} ...
1
vote
1answer
37 views

Limit of sequence and Riemann sum problem work verification

I have to calculate $$\lim_{n \to \infty}{\sum_{k=1}^{n}{\frac{n}{k^2-4n^2}}}$$ My attempt: $$\lim_{n \to \infty}{\sum_{k=1}^{n}{\frac{n}{k^2-4n^2}}} = \lim_{n \to ...
2
votes
1answer
24 views

Integrable function $f$ on $(\mathbb N, \mathcal P(\mathbb N),\mu)$ and series

Problem Let $(\mathbb N, \mathcal P(\mathbb N),\mu)$ where $\mu(A)=card(A)$. Show that $f \in L^1(\mathbb N,\mu)$ if and only if $\sum_{n=1}^{\infty} |f(n)|<\infty$, in which case $\int_X f ...
6
votes
1answer
292 views

Integral of the ratio of two exponential sums

I am trying to find a lower bound on the following integral \begin{align*} \int_{y=-\infty}^{y=\infty} \frac{ (\sum_{n=[-N..N]/\{0\}}n e^{-\frac{(y-cn)^2}{2}})^2} {\sum_{n=[-N..N]/\{0\}} ...
4
votes
3answers
98 views

$ \frac{1}{2} + \dots + \frac{1}{n} \le \log n $

could anyone give me any hint how to prove this ? $$ \frac{1}{2} + \dots + \frac{1}{n} \le \log n $$ just came acroos this expression in my book.
1
vote
2answers
64 views

How do I compute $ \sum\limits_{n=0}^{\infty} \frac{1}{(n+2)7^n} $?

I'm trying to evaluate the following sum: $$ \sum\limits_{n=0}^{\infty} \frac{1}{(n+2)7^n} $$ Probably the best way to do that is to try and find a closed formula for $$ \sum\limits_{n=0}^{\infty} ...