Tagged Questions

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

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0
votes
3answers
85 views

Can I know the value of an infinite serie?

$$\sum\limits_{n=0}^{\infty}\frac{n}{e^n}$$ I have found through a software that the value is $\dfrac{e}{(e-1)^2}$. I've been trying to do it manually but I am getting $\dfrac{\infty}{\infty}$, ...
0
votes
1answer
33 views

If $f_{n-1}(x)=nf_{n}'(x)$ , write $ f_n(x)$ as a function of $f_1(x)$

Let $f_{n-1}(x)=nf_{n}'(x)$ for all $n>1$, and $f_1(x)=a$, what is the expression depicting the relationship between $ f_n(x)$ and $f_1(x)$? I need help with this series.
0
votes
0answers
28 views

If $y_m \to y$ in $H$ then $|y_m|_H \leq C|y|$ for this sequence?

Let $w_i$ be a basis for a Banach space $V$. We have $V \subset H$ a continuous and dense embedding into a Hilbert space $H$. Define $y_m = \sum_{i=1}^m a_{im}w_i$. We have that $y_m \to y$ in a $H$ ...
7
votes
4answers
103 views

Example of an $(a_n)$ sequence with exactly $k$ limit points

It is a well-known result that the sequence $$ a_n= \frac{(-1)^nn}{n+1}, $$ has two limit points, and these are $1$ and $-1$. I'm just looking for some examples of sequences that have exactly $k$ ...
0
votes
1answer
53 views

Does the series $\sum_{n=1}^{\infty} \sin\left((-1)^{n}\over n\right) $ converges conditionally? [closed]

Which convergence tests can I use in order to show that the series $$\sum_{n=1}^{\infty} \sin\left((-1)^{n}\over n\right) $$ converges conditionally.
7
votes
1answer
97 views

A cute limit $\lim_{m\to\infty}\left(\left(\sum_{n=1}^{m}\frac{1}{n}\sum_{k=1}^{n-1}\frac{(-1)^k}{k}\right)+\log(2)H_m\right)$

I'm sure that for many of you this is a limit pretty easy to compute, but my concern here is a bit different, and I'd like to know if I can nicely compute it without using special functions. Do you ...
3
votes
0answers
35 views

Explicit definition from recursive definition

I have the recursive definition: $$a_0=0,~a_{n+1} = 28 + a_n + \left\lfloor\frac{a_n}{16}\right\rfloor$$ I want to create an explicit form for that. I was able to transform the problem into finding an ...
0
votes
2answers
29 views

Number Sequences

These are the terms the question gave me. Term 1 = 1 Term 2 = 1 Term 3 = 2 Term 4 = 3 Term 5 = 5 Term 6 = 8 Term 7 = 13 Term n = ? I found out the pattern which is Term 2 is Term 1 + 0. Term 3 is ...
2
votes
2answers
40 views

Another Summation

After looking at the question here Computing summation I wondered if it might be possible to evaluate the following summation with a similar-looking summand term but with $2n$ instead of $2^n$: ...
0
votes
1answer
67 views

Prove (or disprove) this $\sum_{n=1}^{\infty}\dfrac{a_{n}}{n}$ is convergence? [closed]

Let $a_{n}>0$ be a sequence, and $0<a\le 1$, such that $\sum\limits_{n=1}^{\infty}(a_{n})^a$ converges. Prove or disprove $\sum\limits_{n=1}^{\infty}\dfrac{a_{n}}{n}$ is convergent. I ...
0
votes
1answer
25 views

Convergence of sequence of functions on Banach space

Let $\{f_{\alpha_n}\}\subset{\cal L}_2^0(\mathbb R)$ be a sequence function converging to $g$ where ${\cal L}_2^0(\mathbb R)$ is a Banach space defined by $$ {\cal L}_2^0(\mathbb ...
1
vote
2answers
40 views

Convergence of $\sum_{n} \frac{a_n}{2^n}$ where $a_n\rightarrow +\infty$

Does there exists a sequence $a_n$ of positive reals such that $a_n\rightarrow \infty$ but $\sum_{n}\frac{a_n}{2^n}$ converges? (I considered $a_n=n$, but couldn't prove convergence or divergence of ...
2
votes
3answers
68 views

$\sum (\frac{1-2n}{6+2n})^n $ converges?

Verify if $$\sum_{n=0}^{\infty} \left(\frac{1-2n}{6+2n}\right)^n $$ converges The root test is inconclusive and the limit of the general term is 0. I think I should use the comparison test, in this ...
0
votes
0answers
17 views

Complex form of Fourier series - help

Let function $f(t)$ is represented by Fourier series, $$\frac{a_0}{2}+\sum_{n=1}^{\infty}(a_n\cos{\frac{2n\pi t}{b-a}}+b_n\sin{\frac{2n\pi t}{b-a}}),$$ $$a_0=\frac{2}{b-a}\int_{a}^{b}f(t)dt,$$ ...
2
votes
3answers
82 views

If $a_{n+1}=a_n+\frac1{a_n}$, then $a_n/n$ converges to $0$

Let $a_{n+1}=a_n+\dfrac1{a_n}$, with $a_n=1$. Prove $\lim \limits_{n\to \infty }\left(\dfrac{a_n}{n}\right)=0$. Now I already know that it is monotonically increasing and that $a_n\to \infty$ as ...
3
votes
2answers
46 views

Fractional Sobolev space $H^{1/2}(-\pi,\pi)$

Let $H^{1/2}(-\pi,\pi)$ be the space of $L^2$ functions whose Fourier series coefficients $\{c_n\}_n$ satisfy the summability constraint $\sum_n |n| |c_n|^2 < \infty$. Are functions in ...
0
votes
0answers
42 views

Proving inequality equation

Let $a_1, a_2,....,a_n$ be positive numbers such that $\sum_{i=1}^n a_i = 1$ Then for any vector $(x_1,x_2,...x_n) \ge (0,0,...,0)$ I want to show that $$x_1^{a_1}*x_2^{a_2}*...*x_n^{a_n} \le ...
4
votes
5answers
129 views

Convergence of $\sum \frac{(2n)!}{n!n!}\frac{1}{4^n}$

Does the series $$\sum \frac{(2n)!}{n!n!}\frac{1}{4^n}$$ converges? My attempt: Since the ratio test is inconclusive, my idea is to use the Stirling Approximation for n! $$\frac{(2n)!}{n!n!4^n} ...
0
votes
0answers
30 views

The value of $\sum^{100}_{r=2} \frac{3^r (2-2r)}{(r+1)(r+2)} =$

Problem : The value of $$\sum^{100}_{r=2} \frac{3^r (2-2r)}{(r+1)(r+2)} =$$ Solution : $$\sum^{100}_{r=2} \frac{3^r (2-2r)}{(r+1)(r+2)} $$ Putting r =2,3,4 $\cdots$ $$T_1 = \frac{3}{2}-3$$ ...
2
votes
4answers
62 views

If $1(0!)+3.(1!)+7(2!)+13(3!) +21(4!) + \cdots $ n terms… [closed]

Question( from sequences) : If $1(0!)+3.(1!)+7(2!)+13(3!) +21(4!) + \cdots $ n terms = $(4000)(4000!)$ Then what is the value of n. How to proceed in this please suggest , will be of great help to ...
4
votes
1answer
64 views

Find the sum to n terms of the series

Find the sum to n terms of the series $$\frac {\sin x}{\cos x+\cos2x} + \frac {\sin2x}{\cos x+\cos4x} + \frac {\sin3x}{\cos x + \cos6x} +\dotsb $$ How can I solve this? Here is what I did for the ...
0
votes
1answer
15 views

Function of a sequence, how to answer these types of questions.

I don't quite get this idea of taking the function of a sequence, and what it implies. The questions I am getting go a bit like this; Given a sequence $s_n$ that converges to some $a$ as $n$ gets ...
1
vote
2answers
39 views

What is the sum $\sum_{r=1}^\infty \frac{r}{4r^4+1}$ equal to?

Problem : If $$T_r =\frac{r}{4r^4+1}$$ then the value of $$\sum^{\infty}_{r=1} T_r$$ is ? How to start such problem I am not getting any clue on this please suggest thanks .
3
votes
1answer
26 views

Help explain existence of a limit point of a sequence implies infinitely many $m$ where $d(x,x_m)<\epsilon$

I don't understand the phrase "...all but finitely many elements...". What does this mean exactly and how does the conclusion "Infinitely many elementsof the sequence $\{x_k\}$ must also be within ...
1
vote
2answers
375 views

Convergence of a series 1/(2n+1)

I'm looking for a way to get an estimate on a sum of the following series: $$\sum_{i=1}^{n} \frac{1}{2i-1}$$ My exact question would be the solution for $n=500$ but I'd be interested in the generic ...
6
votes
2answers
70 views

Does $\frac{1}{n}\sum_{i=1}^n|x_i|\to L<\infty$ imply $\frac{1}{n}\max_{1\leq i\leq n}|x_i|=0$?

To simplify notation, let us assume that $\{x_n\}_{n\geq 1}$ is a sequence of nonnegative real numbers. Does $$ \frac{1}{n}\sum_{i=1}^nx_i\to L $$ for some finite $L$ imply $$ \frac{1}{n}\max_{1\leq ...
0
votes
1answer
30 views

Corollary of Kolmogorov zero-one law

Here is another corollary of the theorem: Kolmogorov zero-one law given in my textbook (Probability path). How can I apply the said theorem given that if $X_n$ are independent random variables, ...
1
vote
1answer
32 views

Compactness of the convergent to zero sequences

I've gotta prove that $$T = \left\{ \left\{ x_i \right\} \in {\ell ^\infty }:\left| x_i \right| < \mu_i,\mathop \lim\limits_{i \to \infty } \mu _i = 0 \right\} \subseteq \ell ^\infty $$ is ...
1
vote
1answer
67 views

Subsequences. Prove that (xn) is bounded. Prove that (xn) converges to x.

Let x be a real number, and suppose (xn) is a sequence such that every subsequence (xnk ) of (xn) has a subsequence (xnkj ) that converges to x. (a) Prove that (xn) is bounded. (b) Prove that (xn) ...
1
vote
2answers
52 views

Show that there exists a subsequence $\{F_{n_{k}}\}$ which converges to uniformly on $[a,b]$.

Let $\{f_n\}$ be uniformly bounded sequence of functions which are Riemann-integrable functions on $[a,b]$ and define for $a\leq x\leq b$. $$ F_n(x)= \int_a^x f_n(t)dt.$$ Show that there exists a ...
4
votes
2answers
44 views

Number of Fibonacci series that contain a certain integer

In my question, I consider general Fibonacci sequences (sequences satisfying the recurrence relation $F_{n+2}=F_{n+1}+F_n$ independent of their starting value). Given two arbitrary different integers, ...
3
votes
2answers
109 views

$\int_{0}^{\infty} \frac {\sqrt x}{x^2-5x+6} \,dx = \frac {\pi}{i \sqrt 3 +i \sqrt 2}$ [closed]

Prove $$\int_{0}^{\infty} \frac {\sqrt x}{x^2-5x+6} \,dx = \frac {\pi}{i \sqrt 3 +i \sqrt 2}$$ [Nevermind!]: It has been said, it can be solved in at least three ways.I'm looking forward to seeing two ...
3
votes
3answers
81 views

How can we show that $ \sum_{n=1}^{\infty} \frac{n}{2^n} = 2 $? [duplicate]

How can we prove the following? $$ \sum_{n=1}^{\infty} \frac{n}{2^n} = 2 $$ It would be great to see multiple ways, or hints, about how this can be proven. I know this is a power series ...
5
votes
1answer
169 views

Analyzing this series.

Consider the series $\sum_{n \geq 2} \frac{1}{n^p \ln^qn}$. Prove that: The series converges if $p > 1$ (and any $q$), or if $p = 1$ and $q > 1$. The series diverges if $p < 1$ ...
1
vote
1answer
35 views

Simplify a power series

I am studying bernouli numbers and I'm having trouble condensing a power series. In particular, I'm studying the equation $$b(x)^2=(1-x)b(x)-xb'(x)$$ where ...
3
votes
2answers
57 views

How to show $\sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} = 4 \int_{0}^{\frac{1}{2}} \frac{\arcsin^{2}(x)}{x} \ dx$?

$$\sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} = 4 \int_{0}^{\frac{1}{2}} \frac{\arcsin^{2}(x)}{x} \ dx.$$ Someone please show that this equation is correct !?
3
votes
2answers
65 views

Handling a sequence in a series: $\sum_{n=1}^{\infty} \frac{(a_n)^n \cos(n\pi)}{n}$ for $a_n \rightarrow \tfrac{1}{2}$

The question is whether the series $\sum_{n=1}^{\infty} \frac{(a_n)^n \cos(n\pi)}{n}$, where $\{ a_n \}$ is a sequence of positive numbers that converges to ½, converges absolutely or not. My ...
9
votes
2answers
198 views

Closed form of $\int_0^1(\ln(1-x)\ln(1+x)\ln(x))^2\,dx$

I remember that some time ago I was asking this question Evaluate $\int_0^1\ln(1-x)\ln x\ln(1+x) \mathrm{dx}$ , and now, while I was making a review, I asked myself if we can get the closed form of ...
0
votes
1answer
53 views

Monotone increasing sequence of rationals with an irrational limit

I am trying to use rationals in order to approximate irrationals. Is it possible to construct a monotonically increasing sequence of rationals the limit of which is an irrational? If so, how?
0
votes
2answers
29 views

Derivative of Fourier series

Let function $f(t)$ is represented by Fourier series, $$\frac{a_0}{2}+\sum_{n=1}^{\infty}(a_n\cos{\frac{2n\pi t}{b-a}}+b_n\sin{\frac{2n\pi t}{b-a}}),$$ where $a$ and $b$ are lower and upper boundary, ...
1
vote
1answer
33 views

How to find the sum of $\sum_{n=1}^{\infty} (-1)^{n-1} \frac{3^n}{n5^n}$?

In particular, it's the $\frac{1}{n}$ term that throws me off; without it, it's a simple geometric series, but I was under the impression that, broadly speaking, the only infinite sums whose value we ...
0
votes
3answers
33 views

Test for infinite series convergence or divergence

Hi what would be a good test to find convergence or divergence please? $\sum _1 ^{\infty} (e^kcos^2k)/ \pi ^k$ My attempt I got that converges thanks
-1
votes
1answer
40 views

Finding next number of the logical series [closed]

Find the next element 0 1 1 3 7 31 255 8191 . Also tell the way of finding numbers in such series mathematical
0
votes
1answer
15 views

How to use the triangle inequality to get $d(x_k,y)\ge d(x,y)-d(x_k,x)$ (Prove limit of a sequence is unique)

Here, $d$ is the Euclidean distance: And the triangle inequality in terms of $d$ is: I have no idea how to come up with the $d(x_k,y)\ge d(x,y)-d(x_k,x)$ inequality. What I've tried: ...
-1
votes
0answers
47 views

counting the good numbers

We need to calculate Good Numbers in range from $A$ to $B$ (Both inclusive). A number $N$ is said to be a good Number if it satisfy following conditions : If we extract every $2$-digit number of $N$ ...
-1
votes
2answers
25 views

Convergence/divergence test for infinite integrals

What would be a suitable test for convergence or divergence of the series: $$\sum_{k=1}^{\infty} \frac{k}{k^3+1}?$$
0
votes
1answer
24 views

About integration and the convergence of series of functions

I've tried to solve a problem and there is a question I'm not sure about its answer: What are the conditions for this equality to hold? $\sum_{n=0}^\infty {\int_0^1 f_n(t,x)\ dt} = \int_0^1 ...
4
votes
2answers
68 views

How do I solve this integral? (looking for hints)

I have no idea how to go about solving this integral: $$ \int_0^{\frac{\pi}{2}-}\frac{dx}{1+\tan^{\sqrt{2}}\left(x\right)} .$$ The most I've come up with is rewriting it as a series to obtain $$ ...
1
vote
1answer
42 views

Infinite Products — Tangent function?

I've been looking around and I see no formulas given in any of the sources I've been able to find for the infinite product representing $\tan\left(x\right)$. Is it simply the ratio of the infinite ...
4
votes
1answer
55 views

Computing $\lim_{N\to\infty}\sum_{i = 1}^N \sum_{j = 1}^N \frac{(-1)^{i+j}}{i+j}.$

I'm trying to find $$\lim_{N\to\infty}\sum_{i = 1}^N \sum_{j = 1}^N \frac{(-1)^{i+j}}{i+j}.$$ My attempt was to use a Riemann sum... Thanks for your help.