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

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2
votes
6answers
74 views

How to find the limit $\lim_{n\to \infty}\frac{n}{2^n-1}$?

I've been trying to find and prove the $$\lim_{n\rightarrow\infty}\frac{n}{2^n-1}$$ but I haven't even figure out the limit, could anyone help?
2
votes
1answer
103 views

How to find the general sum formula of this logarithmic series $\log 5+\log 5+ \log 605+\log 6655+\dots$

I have another question about series. Now, this is about series involving logarithm. In the previous post, we can easily grouping the same factor from this series: http://tinyurl.com/kbg26ye But, ...
2
votes
0answers
37 views

Recurrence with Polynomial Coefficients of $n$

How would I solve or obtain a closed-form solution for a recurrence relation like $$a_n = f_1(n)a_{n-1} + f_2(n)a_{n-2} + ... + f_k(n)a_{n-k} + g(n)$$ where $f_1, f_2, ..., f_k, g$ are polynomials and ...
1
vote
1answer
25 views

Power series of $\sum_{n=0}^{\infty}\frac{a_{2n}}{2}z^{2n}$ and $\sum_{n=0}^{\infty}a_nz^n$

Consider the two complex power series $$\sum_{n=0}^{\infty}a_nz^n-(1)$$ $$\sum_{n=0}^{\infty}\frac{a_{2n}}{2}z^{2n}-(2)$$. Say the radius of convergence of (1) is R (finite). What can be said about ...
0
votes
1answer
30 views

Convergence/ divergence tests

If i split this equation to test for convergence/divergence, I get one part to be divergent and the other convergent. Can I say something meaningful about convergent + divergent = ? Or is there ...
4
votes
7answers
905 views

Formula for producing numbers between 0 and 255?

I'm writing a program that needs to cycle through numbers between 0-255 given mouse X position. If given number goes over 255, the difference should be subtracted, as in ...
1
vote
1answer
23 views

$\text{limsup}|a_n|^{1/n}\geq \text{limsup}|a_{2n}|^{1/n} $

The question is very simple. Is it true that $\text{limsup}|a_n|^{1/n}\geq \text{limsup}|a_{2n}|^{1/n} $ where $<a_n>$ is some real sequence? Can this be proved or disproved? And also what if I ...
2
votes
2answers
55 views

Give a general formula for the following sequence

I have a sequence where the terms are $1,0,-1,-1,0,1,1,0,-1,-1...$. Can someone help me find a general formula in terms of $n$ for this sequence?
1
vote
3answers
45 views

Sum to infinity of An and Sk

Consider a series $\sum_{n=1}^\infty A_n$ for which $S_k = \frac{k + 1}{k}$ $\forall k \in \mathbb N$. Find $A_n$ and the value of $\sum_{n=1}^\infty A_n$. I think i have worked out that ...
4
votes
2answers
52 views

Show that $nu_n$ converges to $1$.

$\Bbb{K}=\Bbb{R}$ or $\Bbb{C}$ Let $(u_n)\in\Bbb{K}^{\Bbb{N}}$ be a sequence such that $(n(u_n+u_{2n}))_{n\in \Bbb{N}}$ converges to $\frac{3}{2}$ and $u_n\rightarrow 0$. It is asking to ...
1
vote
1answer
41 views

Let $\tau=\{(-\infty,a)\ :\ a\in\mathbb{R}\}\cup\{\emptyset, \mathbb{R}\}.$Does $\{\frac{1}{n}\}_n$ converge or diverge in this topology.

Let $\tau=\{(-\infty,a)\ :\ a\in\mathbb{R}\}\cup\{\emptyset, \mathbb{R}\}.$ Discuss the convergence or divergence of $\{\frac{1}{n}\}_n$ in this topology. If $\{\frac{1}{n}\}_n$ converges to $p$, ...
-1
votes
1answer
53 views

How to find the formula for these repeating sequences?

How to find formula for this number pattern? 3,6,5,2,3,... When plot this sequence into the graph, it is going to be the sine graph..
0
votes
3answers
63 views

Constructing a recursive sequence that converges to sqrt 17

One of the problems that we have for abstract math is the following: Using the recursive sequence definition, construct a sequence that converges to $\sqrt{17}$. It is my understanding that the ...
0
votes
1answer
34 views

How is this the $b_n$ in this series? (Alternating Series Test)

I am using the Alternating Series Test for this series: $$\sum_{n=0}^{\infty} \frac{\sin(n+1)\pi}{1+\sqrt n}$$ Why is my $b_n$ this?: $$\frac {1}{1+\sqrt n} $$ I dont understand what makes the ...
0
votes
6answers
66 views

For what values of $x$ does the series converge?

For what values of $x$ do the following series converge or diverge $$\sum \limits_{n=1}^{\infty} \frac{x^n}{n^n}$$ I tried to solve this using the ratio test where the series converge when $$\lim ...
2
votes
3answers
43 views

Recursive Sequence solving for $f(200)$

Let $f$ be defined recursively by: $f(0)=5$ and $f(n+1)=3f(n)-2$. Find $f(200)$ I'm really confused how to go about solving this. Can someone help? Thank you!
1
vote
1answer
51 views

Deduce that $e$ is irrational from the following inequality…

Deduce that $e$ is irrational from the following inequality.... $0 < e-\sum\limits_{k=0}^{n}\frac{1}{k!}<\frac{1}{n!n}$ where $n\geq1$ Fairly straightforward to show that $0 < ...
3
votes
0answers
92 views

Proving convergence of series [duplicate]

Prove whether the following series converge or diverge. $$\sum \limits_{n=1}^{\infty} \frac{(2n)!}{(4^n)(n!)^2(n^2)}$$ I think this series converge and I tried to justify using the ratio test but I ...
0
votes
4answers
69 views

Determine the convergence of a series

Prove whether the following series converge or diverge. $$\sum \limits_{n=1}^{\infty} \frac{2^n+3^n}{4^n-5^n}$$ I think this series converge so I tried to prove with ratio test and comparison test ...
0
votes
2answers
42 views

Prove convergence by considering the partial sums

Let $p$ be a non-zero natural number. Prove by considering the partial sums that $\sum \frac{1}{k(k+p)}$ converges. What is $\sum\limits_{k=1}^{\infty} \frac{1}{k(k+p)}$ No idea. Obviously, it ...
1
vote
2answers
35 views

monotonic implication?

let $A_n$ be a monotonic sequence such that $\forall n, A_n \in \mathbb{Z},\,A_n<A_{n+1} $ given the sequence $B_n = (1+{1\over A_n})^{A_n}$ $$\lim \limits_{n \to \infty} B_n = ?$$ now, just ...
0
votes
1answer
39 views

Demonstrate the existence of the following limit

Prove that for $m \geq n \geq 1$ that $|a_m-a_n| \leq n^{-1}$ and deduce that $(a_n)$ converges. For $n\in \mathbb{N}$, denote $$a_n=\int\limits_1^n\frac{\cos(x)}{x^2}dx.$$ By integration by parts, ...
0
votes
1answer
115 views

Using the contraction principle to prove that the sequence $a_n=f(a_{n-1})$ is convergent where $f(x)=1+1/x$

Define function $f(x)$ from $\mathbb R^1$ to $\mathbb R^1$ by $f(x) = 1+1/x$. Define $a_n$ inductively by $a_1=1$ and $a_n=f(a_{n-1})$ for $n \geq 2$. Prove using the contraction principle that ...
0
votes
1answer
121 views

Convergent Series 2n-1/2n

Prove the series defined by P(n) = (1 *3 * 5 * (2n-1))/(2*4*6 * (2n)) is convergent It is monotone decreasing and bounded below by zero, but is that enough to say?
1
vote
1answer
18 views

Can this series be put in a generalized form?

I asked a similar question here But this one seems to not work out so nicely... I started looking at the series, $$ S = ...
24
votes
2answers
545 views

Are there any visual proofs for $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$?

I was flipping through Proofs Without Words (PWW) and saw many visual proofs for sequences and series. However, I saw none for $$\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$$ Are there any ...
3
votes
2answers
44 views

Quadratic Sequences

so i'm trying to find the general formula of a sequence, $${18,30,46,66,90,118}$$ i found the first difference, $${12,16,20,24,24}$$ second difference, $${4,4,4,4}$$ If the second difference is 4, ...
0
votes
1answer
18 views

Taylor's Inequality - What is x?

In Taylor's Inequality, $|R_n(x)| \le \frac{M}{(n+1)!}|x-a|^{n+1}$ What exactly goes inside the $|x-a|$ portion of the equation? The problem I'm given is asking me what would give a bound on the ...
2
votes
4answers
129 views

Discuss the convergence of the sequence whose $n$-th term is given by $a_n= \frac{n!}{n^n}$

Discuss the convergence of the sequence whose $n$-th term is given by $$a_n= \frac{n!}{n^n}$$ I'm just a little confuse on how this can be solved because the factorial is confusing me.
1
vote
1answer
32 views

Find $a_n$ using the partial sum of series

The sequence $(s_k)_{k \in \mathbb{N}}$ of partial sums of a series $\sum _{n=1}^{\infty}a_n$ is defined by $s_k= \sum _{n=1}^{k}a_n$. Consider a series $\sum _{n=1}^{\infty}a_n$ for which $s_k= ...
1
vote
2answers
69 views

Show that $(s_n)$ is a Cauchy sequence

$$S_n = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + ... + \frac{(-1)^{n+1}}{2n-1} $$ Show that $(S_n)$ is a Cauchy sequence and hence that it converges to limit $L$. Show that $\frac{2}{3} < L ...
1
vote
1answer
18 views

Generating sequence

I have these two sequences $ a,b $ $$ \begin{array}{c|lcr} n & \text{a} & \text{b} \\ \hline 1 & 1 & 0 \\ 2 & 2 & 1 \\ 3 & 2 & 0 \\ 4 & 3 & 2 \\ 5 & 3 ...
1
vote
2answers
69 views

Infinite Sequence Converging to x proof

I really don't understand how to do proofs on convergence at ALL. I know you're supposed to use $|xi - x|$ < $\epsilon$ but I have no idea how to apply this to this question: Show that if $x$ is ...
1
vote
2answers
35 views

Finding the limit of a series

I got the n'th term but clueless about what to do next
2
votes
2answers
160 views

Prove that if $\sum a_{n}$ converges, then $ \sum \frac{1}{a_{n}}$ diverges.

Prove that if $\sum a_{n}$ converges, then $ \sum \frac{1}{a_{n}}$ diverges. As $\sum a_{n}$ converges, $a_{n}$ converges to 0. Therefore $\lim \limits_{n \to \infty} \frac{1}{a_{n}} \neq 0$ so ...
1
vote
2answers
34 views

Help to find example where $\sup\{a_nb_n|n\in N\}> \sup\{a_n|n\in N\}\sup\{b_n|n\in N\}$

In the past few hours I'm trying to find an example where $$\sup\{a_nb_n|n\in N\}> \sup\{a_n|n\in N\}\sup\{b_n|n\in N\}$$ But I just can't find it, can someone see what I miss here? EDIT: ...
0
votes
1answer
82 views

Does the series $\sum \sin(100n)$ converge? [duplicate]

Does the following series converge? $$\sum \sin(100n) = \sin(100) + \sin(200) + \dots$$
0
votes
3answers
29 views

If a series converges, then the following sequence converges to $0$

Suppose $\sum_nx_n$ converges to $A$. Then how do I show that $$\lim_{n\to\infty} x_{n+1}+...+x_{n+k}=0?$$ First I need to show that this sequence converges, right? If I can show that the rest is ...
0
votes
2answers
27 views

I wonder how to multiple this sequences?

If I have two sequences: $$a_n=\{n|n\le5,n\in N\}$$ $$b_n=\{n|n\le2,n\in N\}$$ So $a_n$ members are $a_1=1,a_2=2,a_3=3,a_4=4,a_5=5$ and thats it. And $b_n$ members are $b_1=1,b_2=2$ and thats it. So ...
1
vote
1answer
28 views

Limit law proof for max

I am working towards an extremely difficult real analysis problem. The statement is as follows: Prove that if $\lim_{x \to a} f(x) = l$ and $\lim_{x \to a} g(x) = m$ then $\lim_{x \to a} ...
0
votes
1answer
27 views

How to prove that $\sup\{a_nb_n|n\in N\}\le \sup\{a_n|n\in N\}\sup\{b_n|n\in N\}$

It is known that $$a_n,\ b_n\ge0.$$ And they are both upper bounded. Knowing this how can one prove that $$\sup\{a_nb_n|n\in N\}\le \sup\{a_n|n\in N\}\sup\{b_n|n\in N\}$$ I don't see how to approach ...
0
votes
1answer
39 views

Proving that $\exists x\in[0,1]:\sum\frac{b_n^2}{|x-a_n|}<\infty$

Let $\{b_n\}$ be a sequence of positive numbers s.t $\sum b_n<\infty$ and let $\{a_n\}$ be a sequence of real numbers in $[0,1]$. Prove that $\exists ...
2
votes
2answers
43 views

How do I show that the following series converges?

$$\sum_1^\infty \frac{(-1)^n (1 + 1/2 + \cdots+ 1/n)}n$$ I tried applying the alternating series test, but I think that fails. I don't know which other test I could use here.
0
votes
0answers
33 views

Approximation of an infinite series using an integral.

For electric potential I have the following infinite series: $V=k_e\frac{q}{r}+2k_eq\sum_{n=1}^{\infty}\frac{1}{\sqrt{r^2+n^2a^2}}$ Taking the derivative with respect to r, I have the following ...
3
votes
1answer
57 views

$\{1/n\}_n$ converges in the Sorgenfrey line

The following is my attempted proof that the sequence $\{ \frac{1}{n} \}_n$ converges in the Sorgenfrey line. Offer criticism, please! Consider ...
2
votes
1answer
79 views

$\{1-\frac{1}{n} \}_n$ does not converge in the Sorgenfry line

I am trying to prove that the sequence $\{1-\frac{1}{n} \}_n$ does not converge in the Sorgenfry line. Below is my attempt. Consider ...
1
vote
2answers
79 views

What is the correct notation for a sub-sequence?

I want to say that a sequence is a subsequence of $A_n$ for all even indexes, is it valid to write it as $$ k\in \mathbb{N},\, n > k$$ $$(B_k){\,^\infty_{k=1}} = (A_n){\,^\infty_{n=2k}} $$ or $$ ...
1
vote
2answers
60 views

$\sum^{\infty}_{n=1}(n-\sqrt n)/(n^{2}+5n)$ diverges

Show that $$\sum^{\infty}_{n=1}\frac{n-\sqrt n}{n^{2}+5n}$$ diverges. I have tried Root test, ratio Test, Cauchy condensation Test but all have failed. I think this has to be done by Comparison Test ...
2
votes
2answers
101 views

Cesàro summable sequences

During some homeworks the following question came into my mind (it is not part of the homeworks): Let $(a_k)_{k \in \mathbb{N}}$ be a Cesàro summable sequence in $\mathbb{C}$ and let $a := \lim_{n ...
0
votes
2answers
40 views

sum of series independent of variable in sequence

\begin{equation} -\frac{k^2+4}{k^2-8}-\frac{12k^2}{(k^2-8)^2}\sum\limits_{n=0}^{\infty} (\frac{k^2}{k^2-8})^n=\frac{1}{2} , -2\leq k\leq 2 \end{equation} This is an equation which sums up to a value ...