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

learn more… | top users | synonyms (5)

2
votes
2answers
43 views

Infinite sequence of $\sqrt{2^n}$ equals $i$?

So I am no mathematician, in fact I consider myself not very good at math at all, however I do enjoy it. Anyways, I was messing around when I remembered a numberphile video I watched a while back ...
0
votes
1answer
23 views

Time series closed form similar to harmonic series

I want to find the closed form for the following: $$ S(k, \alpha) = \sum_{t=T-k}^T \frac{\alpha^{-t}}{t} $$ when $\alpha \in (0,1)$. For harmonic series there is an easy way to upperbound: $$ H(k) ...
1
vote
4answers
75 views

How to sum $\sum_{n=1}^{\infty} \frac{1}{n^2 + a^2}$?

Does anyone know the general strategy for summing a series of the form: $$\sum_{n=1}^{\infty} \frac{1}{n^2 + a^2},$$ where $a$ is a positive integer? Any hints or ideas would be great!
0
votes
0answers
7 views

Need an explanation for Variational Iteration Method.

I read paper written by Onur Kiymaz and Aysegul Cetinkaya on 'Variational Iteration Method for a Class of Nonlinear Differential Equations'. On the first pages they introduced the method and later ...
1
vote
4answers
65 views

Show that an increasing sequence diverges if and only if it is unbounded.

Show that an increasing sequence diverges if and only if it is unbounded. How should I go about proving this?
1
vote
2answers
48 views

Prove a sequence converges to f(A).

I would like to know if this is an accurate proof
0
votes
1answer
26 views

Stuck on a difference equation which requires an A-level method

In the non-zero sequence $x[n-1]+x[n+1]=ax[n]$ and $x[n+4]=-x[n]$ i) Find possible values of $a$. ii) For what values of $b$ is $b^n$ a solution ($x[n]=b^n$)? I need to solve this using only ...
-4
votes
1answer
39 views

True or False? If true provide a proof. If false provide a counterexample. [closed]

A) Every subset of R has a least upper bound. B) If a sequence is not monotonic then it diverges. C) Let f : A -> B and g : B -> C be functions. g o f is surjective if and only if f and g are both ...
1
vote
5answers
52 views

Prove that the limit as the geometric sequence, with |r| < 1, goes to 0

I know I should know how to do this, but I can't think how: Prove that the $\lim_{n\to \infty} r^n = 0$ for $|r|\lt 1$. I can't think of a sequence to compare this to that'll work. L'Hopital's ...
6
votes
6answers
126 views

Convergence of $\sum_{n=1}^{\infty}\left(\, \frac{1}{n} - \frac{1}{n + 2}\,\right)$

What criteria can I use to prove the convergence of $$ \sum_{n=1}^{\infty}\left(\,{1 \over n} - {1 \over n + 2}\,\right)\ {\large ?} $$ My idea was to use ratio test: $$\displaystyle{1 \over n} - {1 ...
2
votes
1answer
29 views

Convergence of a nice serie

For which value of $a>0$ and $b>0$ does $$\sum_{n\geq 0}\frac{a^n2^{\sqrt{n}}}{2^{\sqrt{n}}+b^n}$$ converge? Obviously it does not when $b<1$, but i don't have any answer otherwise.
0
votes
4answers
49 views

Convergent series with general term $a_nb_n$

I have the following question. Suppose that $\sum_{n=0}^\infty a_n$ is a convergent series, with $a_n > 0$, and suppose that $b_n > 0$ is a bounded sequence. Then show that the series ...
0
votes
1answer
11 views

partial limits of two sequences

let ${a_{n}}$ ${b_{n}}$ be sequences. $\lim_{n\to \infty}(a_{n}-b_{n})=0$ prove that both seqneces' subsequences' limits are exactly same. right now I have no clue where should I begin. any ...
0
votes
1answer
26 views

Prove the following about absolutely convergent complex series

Prove that for every sequence $(a_n)_n$ of complex numbers, if the series $\sum_{n\ge 0} a_n$ is absolutely convergent, then $|\sum_{n\ge 0} a_n| \le \sum_{n \ge 0} |a_n|$. I've been given the ...
2
votes
1answer
27 views

If $a_n = \sum^n_{r=0} \frac{(\ln10)^n}{r! (n-r)!}$ for $n \geq 0$ …

Problem: If $a_n =\sum^n_{r=0} \frac{(\ln10)^n}{r! (n-r)!}$ for $n \geq 0$ then find the value of $a_0+a_1+a_2+\cdots \infty$ My approach: $a_n = \sum^n_{r=0} \frac{(\ln10)^n}{r! (n-r)!}$ $= ...
3
votes
3answers
151 views

Computing the infinite sum, $\sum_{n=0}^\infty \frac {5^n}{25^n + 1} $

So I'm trying to compute $$ \displaystyle\sum_{n=0}^\infty \dfrac {5^n}{25^n + 1}. $$ The closed form is not very nice and I don't see any immediate telescoping. Any ideas?
3
votes
2answers
63 views

Calculate $\sum_{j=0}^k\binom {2k+1}{2j+1}^2=?$

Knowing that: $${2k\choose k}=\sum_{j=0}^k{k\choose j}^2.$$ calculate the sums: $$\sum_{j=0}^k\binom {2k+1}{2j+1}^2=?$$ Any sugestions please? Thanks in advance.
1
vote
2answers
43 views

Proof limit of sequence with square root [closed]

We know that $\displaystyle\lim_{n \to \infty}{\sqrt[n]{a_n}} = 1$ and $\displaystyle\lim_{n \to \infty}{\frac{b_n}{a_n}} = g$ where $g \in (0, +\infty)$. I have to show that $\displaystyle\lim_{n \to ...
3
votes
3answers
120 views

A sequence of roots of polynomials depending on an integer parameter

For $n\in \mathbb N-\{0\}$, let $$Q_n=(2n-1)X^n+(2n-3)X^{n-1}+(2n-5)X^{n-2}+\cdots+3X^2+X$$ I want to show that there is a unique $x_n\geq 0$ such that $Q_n(x_n)=1$ and then show that the sequence ...
1
vote
1answer
24 views

Limit of recursive sequence

Given non-zero sequence $r_n$ such $\lim_{n \to \infty}{r_{2n}} = \frac{1}{2008}$ and $\lim_{n \to \infty}{r_{2n + 1}} = \frac{1}{2009}$ and sequence $a_n$ such $a_1 = x$ and $a_{n+1} = a_n \cdot r_n$ ...
0
votes
2answers
31 views

Proof that product of series converges [on hold]

$\sum_{n=1}^{\infty}{b_n}$ converges and $\sum_{n=1}^{\infty}{a_n}$ converges absolutely. How to show that $\sum_{n=1}^{\infty}{b_n \cdot a_n}$ converges absolutely? If $\sum_{n=1}^{\infty}{a_n}$ ...
0
votes
2answers
24 views

Find an example of series

Can someone show an example of series that a)$\sum_{n=1}^{\infty}{a_n}$ is conditionally convergent such $\forall_{n \in \mathbb{N}}{|a_n|<\frac{1}{n}}$ b)$\sum_{n=1}^{\infty}{a_n}$ that diverges ...
2
votes
0answers
46 views

Is it true that if $\lim a_n = g$ then $\lim |a_n| = |g|$?

The question is in the title. I'm trying to prove that if $\lim_{n\rightarrow\infty} a_n = g$ then $\lim_{n\rightarrow\infty} |a_n| = |g|$. Is the following proof correct? If ...
3
votes
2answers
89 views

Find the function of integer numbers $\sum_{n=0}^{\infty }\frac{n^k}{n!}=f(k) \cdot e$

Find the function of integer numbers $$\sum_{n=0}^{\infty }\frac{n^k}{n!}={f(k)}\cdot e$$ I took many values of $k$ and I found the following results $$\sum_{n=0}^{\infty }\frac{n^1}{n!}=e$$ ...
2
votes
1answer
35 views

How to evaluate this series? [duplicate]

I need to evaluate this series.Without using derivative. $$A=\frac{2} {2^2} + \frac{4} {2^5}+ \frac{6} {2^8} + \cdots $$ Where the $i$ th member is calculated with the the formula below: ...
3
votes
1answer
43 views

How can I prove $\sum_{n=0}^{\infty }\frac{a.n-(a-1)}{n!}=e$

How can I prove this $$\sum_{n=0}^{\infty }\frac{a*n-(a-1)}{n!}=e$$ When $a$ is any real number
-1
votes
0answers
29 views

Does the sequence: $f_n(x) = (1 − \frac{1}{n})x$ for $x \geq 0$ and $f(x) = −1$ for $x < 0$, converge uniformly? [closed]

Let $$f_n(x) = \begin{cases} (1 − \frac{1}{n})x & x \ge 0, \\ -1 & x < 0. \end{cases}$$ Find the limit function $f(x)$ such that $f_n(x) \to f(x)$ on $\mathbb{R}$. Does $f_n$ converge to ...
2
votes
0answers
42 views

Is $f$ continuous if for every $p$, there is a sequence $p_n \to p$ such that $f(p_n) \to f(p)$?

Let $(X, d)$ be a metric space and $f : X \rightarrow X$ a function that satisfies the following property: For every $p \in X$ there exists a sequence $\{p_n\}\subset X$ such that $p_n \rightarrow p ...
0
votes
2answers
15 views

There is no increasing positive sequence $(u_{n})_{n}$ with this condition: $(u_{n}^{(1/n)})_{n}$ is decreasing

Let $(u_{n})_{n}$ be an increasing positive sequence. My question is about this claim (maybe it is false): There is no increasing positive sequence $(u_{n})_{n}$ with this condition: ...
0
votes
1answer
59 views

How to construct a subsequence that is not the sequence with this specific critera and increasing indices? [on hold]

Say that $x_n \nrightarrow x$. I want to create a subsequence $x_{n_k}$ so that this subsequence has a subsequence $x_{n_{k_j}} \nrightarrow x$. (Reminder for non-convergence: If $x_n \nrightarrow x$, ...
8
votes
2answers
152 views

Formula for a periodic sequence of 1s and -1s with period 5

I've been playing with periodic sequences of 1s and -1s lately. This is what I came up with: \begin{eqnarray*} -(-1)^n& = &1, -1, 1, -1,\ldots\quad\textrm{(Period 2)}\\ ...
-1
votes
1answer
32 views

Can I prove these series with limit a using induction?

This is the equation: It is true for: E a normed space and $(a_n)_{n \in \mathbb N}$ a convergent sequence with limes a. $$s_k = \frac1k\sum^k_{n=1} a_n \rightarrow a$$ $a = \lim_{n\rightarrow ...
2
votes
1answer
14 views

Recursive sequence with Complex numbers, missing conclusion.

I am solving the following task: Let $a_1 = \sqrt{2}\sqrt{3}*i, a_{n+1} =\frac{i* a_n}{n+1}$ What can you say about the convergence of $a_n$? I already found out a lot. What i concluded so far, is: ...
0
votes
1answer
34 views

Dynamical System , Series : can't find the general terms

I have a dynamical system defined as follow : $$V_{n+3} - 6V_{n+2} +12V_{n+1} - 8V_n = 8, ~ \mbox{with}~ V_0=V_1=V_2=1$$ I have to find $V_n$ = ? So I began by solving this equation : $$x^3 -6x^2 ...
4
votes
0answers
107 views
+50

How to prove there exists $n_{1}a_{n_{0}}+n_{2}a_{n_{1}}+\cdots+n_{k}a_{n_{k-1}}<3(a_{1}+a_{2}+\cdots+a_{N})$

Let $a_{1},a_{2},\cdots,a_{N}$ be nonnegative reals, not all $0$. Prove that there exists a sequence $$1=n_{0}<n_{1}<\cdots<n_{k}=N+1$$ of integers such that ...
2
votes
1answer
35 views

limit of series paradoxe?

I'm little bit confused about limit arithmetic: $$\lim _{n\to \infty }\left(1\right)\:=\:\lim _{n\to \infty }\left(n\cdot \frac{1}{n}\right)\:=\:\lim _{n\to \infty ...
4
votes
4answers
100 views

Calculate the result of the following sequence.

I'm stuck with this sequence.I can't calculate the result. Any help would be greatly appreciated. $$A=\frac{1}{2} + \frac{2}{4} + \frac{3}{16} + \frac{4}{32} + ... $$ Please feel free to edit the ...
5
votes
5answers
564 views

Can a sequence which decays more slowly converge?

In Bergman's companion notes to Rudin, he says that "If a sequence of positive terms has convergent sum, so does every sequence of positive terms which decays more rapidly." So given a sequence ...
1
vote
1answer
27 views

How to determine the convergence of this series $\sum_{n=0}^\infty \frac{(2k-1)!!}{(2k+1)\cdot (2k)!!}$

Consider $$\sum_{n=0}^\infty \frac{(2k-1)!!}{(2k+1)\cdot (2k)!!}.$$ Here $(2k)!!=(2k)(2k-2)\cdots 2$. If we use Hadamard test, then the ratio is $1$. How can we determine its convergence and find ...
1
vote
1answer
43 views

Finite natural summation that leads to double exponential results

We know that $$f(n)=\sum_{i=0}^n\binom{n}{i}=2^n$$ and $$g(n)=\sum_{i=0}^ni\binom{n}{i}=n2^{n-1}.$$ Are there any finite natural sums that lead to $2^{2^n}$ or $2^n2^{2^{n-1}}$ results other than ...
2
votes
4answers
45 views

A finite binomial sum

Is their an exact expression for the following sequence involving binomial coefficients $$\sum_{i=0}^n i\binom{n}{i}?$$
0
votes
1answer
22 views

Maclaurin series of $f(x)=x^3\sin 2x$

I need help finding that maclaurin series for following function. $$f(x)= x^3 \sin2x$$ How do you get to the maclaurin series?
-1
votes
0answers
49 views

In $\triangle ABC$ , find the value of $\cos A+\cos B$ [closed]

The sides of $\triangle$ABC are in Arithmetic Progression (order being $a$, $b$, $c$) and satisfy $\dfrac{2!}{1!9!}+\dfrac{2!}{3!7!}+\dfrac{1}{5!5!}=\dfrac{8^a}{2b!}$, Then prove that the value of ...
2
votes
2answers
39 views

Prove the recurrence sequence $a_{n+1} = \frac{4 + 3a_{n}}{3 + 2a_{n}} (a_0 = 1)$ to be bounded by $0$ and $\sqrt{2}$

The sequence is defined by: $a_0 = 1$ and $a_{n+1} = \frac{4 + 3a_n}{3 +2a_n}$. I have to prove that $0 < a_n < \sqrt{2}\,$ holds for any $n \in \mathbb{N}$. My attempt is to assume $0 < ...
1
vote
1answer
31 views

Series Solution to Differential Equation

Given the series $$1 + \sum_{k = 1}^{\infty} \frac{\beta(\beta - 1) ... (\beta - k + 1)}{k!} x^k$$ how can I find a differential equation for which this series is a solution? I don't have any idea ...
2
votes
3answers
109 views

If $\sum x^6_n$ converges, then $\sum x^7_n$ converges too

If $\displaystyle\sum x^6_n$ converges, then $\displaystyle\sum x^7_n$ converges too.
2
votes
1answer
35 views

Need to know why $\sum_{k=0}^{\infty}kr^{k} = \frac{r}{(1-r)^{2}}$

Working on a Stat problem where I must find $E(x)$ of $f(x)=\left(\frac{1}{2}\right)^{x+1}$ for $x=0,1,2,\cdots$ I have, ...
2
votes
2answers
35 views

Harmonic Series question about convergence

For large enough $n \in \mathbb{N}$, consider the sequence $(a_i (n))_{i \in \mathbb{N}} \overset{\Delta}{=} (a_i)_i : a_i(n) \overset{\Delta}{=} a_i = \frac{\sum_{x=1}^n (\frac{1}{x^i})}{i} \forall i ...
2
votes
1answer
39 views

Convergence of $\sum_{n=0}^{\infty} \frac{3^{2n}}{(2n)!}$

I need help determining what following series converges to using the ratio test. $$\sum_{n=0}^{\infty} \frac{3^{2n}}{(2n)!}$$ It's the end that really has me confused with what to do with the ...
1
vote
2answers
44 views

Question on the sum $\sum_{n=1}^{\infty}\frac{x^n}{n} = -\ln(1-x)$

$f(x) = \displaystyle\sum_{n=1}^{\infty}\frac{x^n}{n} = x + \frac{x^2}{2} + \frac{x^3}{3} + ... = -\ln(1-x)$ for $|x| < 1$. $f'(x) = \displaystyle\sum_{n=1}^{\infty}x^{n-1} = 1 + x + x^2 + x^3 ...