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

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0
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1answer
53 views

Suppose $a_n$ converges and $b_n$ divergent. What can you tell about $a_n+b_n$? [closed]

Suppose $a_n$ converges and $b_n$ divergent. What can you tell about $(a_n+b_n)$?
1
vote
1answer
56 views

Got stuck with the leibniz criterion

$\sum_{n=1}^{\infty} \Big(\frac{1}{n^2}+ \frac{(-1)^n}{n} \Big)$ Does the progression converge, absolutely converge or diverge. I tried it with the Leibniz-criterion, but I dont know how to proof ...
3
votes
1answer
141 views

Sum of Infinite Series $1 + 1/2 + 1/4 + 1/16 + \cdots$

Everyone knows about the classic $$ \sum_{i=1}^{\infty} \dfrac{1}{2^i} = 1 $$ However, is there any way to find $$ \sum_{i=0}^{\infty} \dfrac{1}{2^{2^i}} = \dfrac12 + \dfrac14 + \dfrac{1}{16} + ...
2
votes
1answer
50 views

Limit of a sum of roots proof

Given the sequence: $$a_n=\alpha\sqrt{n+a}+\beta\sqrt{n+b}\ with\ \ \alpha,\beta,a,b\in\mathbb{R}\ and\ \alpha,\beta\neq0$$ Prove that $$\lim_{ n\to \infty} a_n = 0\ iff\ \alpha=-\beta$$ I start ...
2
votes
2answers
56 views

Dual of a sequence

Let $S$ be the set of all sequences $(a_1,a_2,\ldots)$ of non-negative integers such that (i) $a_1 \ge a_2 \ge \ldots;$ and (ii) there exists a positive integer $N$ such that $a_n=0$ for all $n \ge ...
2
votes
1answer
75 views

sum of an alternating series

How to evaluate the series below ? $$ \sum_{n=0}^{\infty}\left(-1\right)^{n}\,{2n+1 \over \left(2n+1\right)^{2} + x^{2}} $$ Can we reexpress it in term of an elementary function ?.$\,$ By the way, ...
2
votes
0answers
37 views

The series of reciprocals of the integers that do not contain 9 in their decimal representation

Does the following series converge or diverge? $\sum_{n=1}^{\infty} a_n$ where $a_n = \frac 1 b_n$, and $(b_n)_n$ is the subsequence of $(n)_n$ whose terms do not have a $9$ in their decimal ...
5
votes
4answers
85 views

Convergence of $\sum_{n=0}^{\infty}(-1)^n \frac{2+(-1)^n}{n+1}$

I have to show that the following series convergences: $$\sum_{n=0}^{\infty}(-1)^n \frac{2+(-1)^n}{n+1}$$ I have tried the following: The alternating series test cannot be applied, since ...
1
vote
1answer
43 views

Average of sequence random variables

Let $X_1, X_2, X_3, \dots$ be a sequence of random variables that converges almost surely $$(X_n) \rightarrow X$$ to a number $X \in \mathbb R$ (or more precisely the delta dirac distribution ...
1
vote
2answers
161 views

How to prove, by induction, that an infinitely nested radical is increasing

How do I prove using induction that an infinitely nested radical, like sqrt(1+sqrt(1+sqrt(1+... is increasing. I have seen there are many examples on here like this but haven't seen one that proves ...
4
votes
2answers
114 views

Find $L=\lim \limits_{n\to \infty}\sqrt[n]{x^n+y^n+z^n}$

Find the limit following: $$L=\lim \limits_{n\to \infty}\sqrt[n]{x^n+y^n+z^n}$$ With $x,\: y\: z\in R$ P.S I think this limit result is $L=max\left\{x,\: y\: z \right\}$. But i'm not find it, so ...
1
vote
1answer
174 views

Function iteration and intervals of attraction for fixed points

I am currently studying iteration sequences and I am a bit hung up on one specific bit which involves determining intervals of attraction of fixed points. I've been given a graphical method to ...
2
votes
2answers
68 views

Convergence: $\sum\frac1{n\ln(n^3)}$

How do you test the convergence of $$\sum_{n=2}^\infty\frac1{n\ln(n^3)}$$ I tried using limit comparison test, but had no conclusion.
1
vote
1answer
69 views

Sequence with a contraction mapping of the sum

Consider a continuous function $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ with the following property. There exists $c \in (0,1)$ such that for all $x,y \in \mathbb{R}^n$ it holds that $\left\| f(x) - ...
1
vote
2answers
51 views

Recursive sequence of functions

Recursive sequence of functions: $f_{n+1}= \sqrt{x+f_n}$, $f_1(x)= \sqrt{x}$. this sequence is monotonic, but what is bounding it? thanks
16
votes
0answers
309 views

Find the limit $L=\lim_{n\to \infty} \sqrt{\frac{1}{2}+\sqrt[3]{\frac{1}{3}+\cdots+\sqrt[n]{\frac{1}{n}}}}$

Find the limit following: $$L=\lim_{n\to \infty} \sqrt{\frac{1}{2}+\sqrt[3]{\frac{1}{3}+\cdots+\sqrt[n]{\frac{1}{n}}}}$$ P.S I tried find this limit, but it's made me confused.
1
vote
1answer
85 views

Minimum of a sum

I have the function $$f(x)= \sum_{i=1}^n (x-a_i)^2 \ x\in R$$ I am asked to find the minimum of it. I am lost so any help would be nice. Thanks in advance!
0
votes
1answer
70 views

Let $\{a_n\}$ be a positive monotonic decreasing sequence of real numbers. Show $\lim_{k \rightarrow \infty} \frac 1 k \sum_{n=1}^k a_n = \inf a_n$

Let $\{a_n\}_{n=1}^{\infty}$ be a monotonic decreasing sequence of positive real numbers. Show that $\lim_{k \rightarrow \infty} \frac 1 k \sum_{n=1}^k a_n = \inf a_n$ Suppose we take the sum of the ...
4
votes
4answers
120 views

Evaluate $\frac{1}{3}+\frac{1}{4}\frac{1}{2!}+\frac{1}{5}\frac{1}{3!}+\dots$

Question is to Evaluate : $$\frac{1}{3}+\frac{1}{4}\frac{1}{2!}+\frac{1}{5}\frac{1}{3!}+\dots$$ what all i could do is : ...
1
vote
1answer
124 views

Finding the common ratio of a geometric series from the sum and first term

If the sum of a geometric series is 80, and the first term is 5, and the number of terms is 5, how can I determine the common ratio?
0
votes
3answers
290 views

Does this sequence converge or diverge? $a_n = (-1)^n\frac{n^2+n+2}{2n^2+3n+4}$

We had this question on an exam and I don't know if I got it right or not. We were asked to justify our answer. I said the sequence diverges since the limit bounces between $-\frac{1}{2}$ and ...
0
votes
1answer
42 views

Does series of $\frac{1}{n} (\frac{2}{(-1)^n - 3})^n$ converge?

Does $\sum\limits_{n=1}^\infty \frac{1}{n} (\frac{2}{(-1)^n - 3})^n$ converge? If so, how? I think it converges, but I don't know how to use the alternating series test here, since I can't figure ...
1
vote
2answers
44 views

Generalizing a series of numbers

I know that I studied this long ago but I can't seem to bring the information to mind. I am looking to construct a general formula for the following sequence of numbers -> when $x=4$ : ...
0
votes
1answer
108 views

For a sequence of non negative numbers, if the series converges, then the series of the sequence raised to p also converges if p>=1

Let $p \geqslant 1$ and let $(a_n)$ be a sequence of non-negative numbers. Then if $\sum\limits_{n=1}^\infty a_n$ converges, so does $\sum\limits_{n=1}^\infty a_n^p$. Prove this statement. Sorry, we ...
4
votes
2answers
211 views

How does the Herglotz trick work?

There is a series representation as partial fraction expansion where just translated reciprocal functions are summed up, such that the poles of the cotangent function and the reciprocal functions ...
2
votes
1answer
46 views

What is the limsup of $\sum \limits _{n=1}^\infty \frac{1+\sin{n}}{4}$?

$$\sum \limits _{n=1}^\infty \frac{1+\sin{n}}{4}$$ I computed the first $70$ terms of this series. They are each between $0$ and $1$, but they jump around quite a bit and I can't seem to determine ...
0
votes
2answers
548 views

Proof that a function sequence does not converge uniformly

Suppose we have a sequence of functions defined as : $f_n(t) = \cases{0 & \text{for } t\lt0\cr t^n & \text{for } 0\le t\le1\cr 1 & \text{for } t\gt1\cr ...
4
votes
1answer
68 views

Summation of a floored square root

I am working on a little something and have hit a roadblock of sorts. I have arrived at this equation:$$\sum_{n=1}^{r}\left\lfloor\sqrt{2nr-{n}^{2}}\right\rfloor$$ I am attempting to find some way of ...
1
vote
2answers
54 views

Did I solve this problem correctly? $\sum_{n=1}^{\infty}(-1)^{n+1}\frac{n^2}{n^3+4}$

For the following series: $\sum_{n=1}^{\infty}(-1)^{n+1}\frac{n^2}{n^3+4}$ I found $b_n$ to be $b_n=\frac{n^2}{n^3+4} \gt 0$ and then I took the limit of this and found it to be zero.
0
votes
1answer
53 views

About convergence in Sobolev space

I have $u_{n}$ sequence of $H^{1}_{0}(\Omega)$ where $\Omega$ is open bounded and connected domain in $\mathbb{R}^{n}$ with $n>1$. $u_{n}\rightarrow u$ in $H^{1}_{0}(\Omega)$ norm. Let ...
3
votes
3answers
78 views

Convergence of $\sum _{k=1}^\infty (1-\frac{1}{k})^{k^2}$

Found the alternative form: $\sum _{k=1}^\infty ((1-\frac{1}{k})^{k})^k$. Tried various criteria, no luck so far.
0
votes
2answers
39 views

Help with $\sum_{n=1}^{\infty}\frac{(2n+1)^n}{n^{2n}}$

Not sure how to really get started with this because it seems like a geometric series. Can some provide me a hint as to how I should approach this series? I started to use the ratio test but quickly ...
2
votes
3answers
85 views

On the monotonicity of the sequence $\frac{1}{\log^2 n}$

How the monotonicity of the following sequence: $$ a_n=\frac{1}{\log^2n} $$can be shown? Knowing that $\log n$ is increasing, I thought that I could use the inequality $\log(n+1)>\log(n)$ ...
1
vote
1answer
69 views

Comparison test for series convergence

I have these two series and I've been trying to figure out correct series to compare them to, to prove their convergence. The first one:$$\sum_{n=0}^{\infty} (\cos n \pi) \left( ...
2
votes
1answer
39 views

Convergence conditions of a series

Convergence conditions of: $\displaystyle\sum \frac{\sin(x^n)}{(x+1)^n}$ What i did: $\sum \frac{\sin(x^n)}{(x+1)^n} < \sum \frac{x^n}{(x+1)^n}$ And after I studied with the root test when ...
0
votes
1answer
117 views

How do you find the interval of convergence for the series f(x)=Sin(2x)?

I know that $$\sin(2x)=\sum_{n=0}^\infty{(-1)^n\frac{2^{2n+1}x^{2n+1}}{(2n+1)!}}$$ I did the ratio test and came up with the $$\lim_{n\rightarrow \infty}{\frac{-4x^2}{2(n+1)(2n+3)}}$$
2
votes
1answer
58 views

Convergence of $\sum_{ k\in\mathbb{N} } \left( \frac{\lambda^k}{k!} \right)^n$

We know that $\sum_{ k\in\mathbb{N} } \frac{\lambda^k}{k!} = e^\lambda$. I'm interested in the convergence of $$S^{(n)}=\sum_{ k\in\mathbb{N} } \left( \frac{\lambda^k}{k!} \right)^n $$ for some value ...
1
vote
1answer
42 views

Let G be the set of integer sequences $(a_i)_{i=1}^∞$ for which there is some $N ∈ ℕ$ such that $a_i$ = 0 for all i > N .

Let G be the set of integer sequences $(a_i)_{i=1}^∞$ for which there is some $N ∈ ℕ$ such that $a_i$ = 0 for all i > N . (a) Show that G is an abelian group under the operation: $(a_i)_{i=1}^∞$ + ...
3
votes
0answers
79 views

Interchange of finite sum with convergence sequences.

Hi everyone I'm wondering if the following proof is correct (to be honest at the beginning I have some troubles to understand what the sequences of the sums of convergent sequences has to be, but I ...
4
votes
3answers
75 views

Does the series $\sum \limits _{n=1}^\infty \frac{1}{n} \left[{\frac{1}{(-1)^n-5}}\right]^{n}$ converge or diverge?

$$\sum \limits _{n=1}^\infty \frac{1}{n} \left[{\frac{1}{(-1)^n-5}}\right]^{n}$$ I applied the root test and believe that this series converges. That is, I found that: $$\text{limsup} ...
1
vote
1answer
80 views

Are there a link between series convergence and countability of sets?

Could you please help me understand this question: Suppose $E \subset [0,1]$ and for each sequence $(a_n)$ , $a_n \in E$ and there are no duplicate members at $a_n$ , the series $\sum_{n=1}^\infty ...
2
votes
2answers
67 views

Boundedness of a real sequence.

Let $\{a_n\}$ be a sequence in $\mathbb{R}$ such that $\sum |a_n||x_n| < \infty$ whenever $\sum |x_n| < \infty$. Prove that $\{a_n\}$ is bounded. My Attempt : We have to show $\exists$ $B \in ...
0
votes
3answers
56 views

Is $\sum_{n=1}^{\infty}\frac{2^nn!}{(n+1)!}$ absolutely convergent?

I'm very uncomfortable with factorials just because I haven't done many of them. But my basic understanding is if I start with (for example) $(n+1)!$ then this is equivalent to $(n+1)*(n)$ and if it ...
3
votes
1answer
73 views

Evaluate $ 1 + \frac{1}{3}\frac{1}{4}+\frac{1}{5}\frac{1}{4^2}+\frac{1}{7}\frac{1}{4^3}+\dots$

Evaluate $$ 1 + \frac{1}{3}\frac{1}{4}+\frac{1}{5}\frac{1}{4^2}+\frac{1}{7}\frac{1}{4^3}+\dots$$ All i could do was to see that ...
0
votes
2answers
88 views

Prove sequence in a complete metric space converges if the series of distances converges.

Assume that X is complete, and let $(p_n)$ be a sequence in X. Assume that $\sum\limits_{n=1}^\infty d(p_n, p_{n+1})$ converges. Prove that $(p_n)$ converges. (X,d) is a metric space. Don't quite ...
0
votes
2answers
150 views

Removing the square root in $\sum_{n=1}^{\infty}\frac{\sqrt{n+3}}{4n^2+n+4}$

I know I can use the comparison test to examine this series, but say I wanted to take the limit of this series, how do I handle square root in the series? ...
0
votes
0answers
21 views

If C is a non-empty closed set in $\mathbb{R}^{N}$ and $x \in \mathbb{R}^{N}$, prove there is a point C closest to z

The hint I've been given is to choose a sequence of points in C whose distances from x converge to the infimum. And if C is closed and bounded, how can I prove there is a point of C furthest from x? ...
1
vote
2answers
37 views

How to find the sum of subsequences and sum of the series for series like these?

$$\sum_{n=0}^\infty \frac{3^n+4^n}{(-7)^n}$$ I've been trying to figure it out, but to no avail. I believe it can be simplified to something where I can find the limit, but I cannot figure out how. ...
2
votes
4answers
160 views

Why arithmetic series never sums to a fraction

Sometimes with series we find a solution in a form of a fraction which does not a priori obviously take only integer values. On the other hand from the sum it is pretty obvious that the sequence of ...
5
votes
3answers
136 views

Compute the limit $\lim\limits_{n \to \infty} \dfrac{n!}{n^n}$

I am trying to calculate the following limit without Stirling's relation. \begin{equation} \lim_{n \to \infty} \dfrac{n!}{n^n} \end{equation} I tried every trick I know but nothing works. Thank you ...