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

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0
votes
1answer
30 views

Convergence of $ \sum_{n=1} ^\infty \frac {1}{n(1+\frac {1} {2}+\frac {1} {3}+ \cdots+\frac {1} {n} )}$

Convergence of $$ \sum_{n=1} ^\infty \dfrac {1}{n(1+\frac {1} {2}+\frac {1} {3}+ \cdots+\frac {1} {n} )}$$ Attempt: I believe not a nice attempt: $ n(1+\frac {1} {2}+\frac {1} {3}+ \cdots+\frac {1} ...
-3
votes
1answer
15 views

…is the closed form for sequence A_n. Find c using the Fibonacci and Lucas number sequences. [on hold]

Let $$\begin{align*} A_0 &= 6 \\ A_1 &= 5 \\ A_n &= A_{n - 1} + A_{n - 2} \; \textrm{for} \; n \geq 2. \end{align*}$$ There is a unique ordered pair $(c,d)$ such that $c\phi^n + ...
3
votes
0answers
21 views

Is it possible to eliminate the inner sum to evaluate numerically?

Any hints on how to simplify the following double sum to be able to find the sum at least numerically? $$\sum_{n=2}^{\infty}\frac1{n(n^2-1)} \sum_{k=1}^\infty \frac{(k-1/n)^{2n-2}}{(k+1/n)^{2n+2}}$$ ...
2
votes
5answers
50 views

Convergence of $\sum_{n=1}^{\infty} \log~ ( n ~\sin \frac {1 }{ n })$

Convergence of $$\sum_{n=1}^{\infty} \log~ ( n ~\sin \dfrac {1 }{ n })$$ Attempt: Initial Check : $\lim_{n \rightarrow \infty } \log~ ( n ~\sin \dfrac {1 }{ n }) = 0$ $\log~ ( n ~\sin \dfrac ...
-3
votes
0answers
33 views

How to calculate $\sum_{x=1}^n (n/x)^2$ [on hold]

How does one calculate $$\sum_{x=1}^n \left(\frac{n}{x}\right)^2\ ?$$
2
votes
2answers
37 views

Why is $\lim_{n \to +\infty }{\sqrt[n]{a_1 a_2 \ldots a_n}} = \lim_{n \to +\infty}{a_n}$

Solving some problems regarding limits and sequence convergence, i stumbled upon a task, and it's solution relies on, and i quote: "We now use a well-known theorem : $$\lim_{n \to +\infty ...
-6
votes
1answer
22 views

Question of sequence [on hold]

Write the first and second differences of the sequence: 5,8,11,14,...
0
votes
1answer
20 views

if we know that the n'th term of a sequence is equal to $\frac{7^n - a^n}{7^n}$, does that imply we have a formula for every $n$

If we know that the n'th term of a sum of a geometric sequence is $\frac{7^n - a^n}{7^n}$ where $a > 0$, does that mean we have a formula for finding any member of the sum of a geometric sequence ...
1
vote
3answers
48 views

convergence of $ \sum_{n=1}^\infty \frac {1}{\log (1 +\frac {1}{n})}$

Test convergence of $$ \sum_{n=2}^\infty \dfrac {1}{\log (1 +\frac {1}{n})}$$ I am not really sure how to move forward. Could anyone give me a direction to proceed please. EDIT" The only part I ...
5
votes
0answers
38 views

Evaluate $S=\left|\sum_{n=1}^{\infty} \frac{\sin n}{i^n \cdot n}\right|$

Evaluate $$ S=\left|\sum_{n=1}^{\infty} \dfrac{\sin n}{i^n \cdot n}\right|$$ where $i=\sqrt{-1}$ For this question, I did the following, Let $$ \begin{align*} S &= \sum_{n=1}^{\infty} ...
-2
votes
1answer
46 views

A closed form for the sum $S = \frac {2}{3+1} + \frac {2^2}{3^2+1} + \cdots + \frac {2^{n+1}}{3^{2^n}+1}$ is … [on hold]

A closed form for the sum $S = \frac {2}{3+1} + \frac {2^2}{3^2+1} + \cdots + \frac {2^{n+1}}{3^{2^n}+1}$ is $1 - \frac{a^{n+b}}{3^{2^{n+c}}-1}$, where $a$, $b$, and $c$ are integers. Find $a+b+c$.
2
votes
3answers
59 views

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

Convergence of $$\sum_{n=1}^{\infty} \dfrac {1}{n\log^2(n+1)}$$ Attempt: We note that $\lim_{n\rightarrow \infty} \dfrac {n}{ \log^2(n+1)} = \infty$ Hence, for a sufficiently large $n: \dfrac {n}{ ...
0
votes
2answers
55 views

convergence of $\sum_{n=1}^{\infty} \frac {1}{\log(e^n+e^{-n})}$?

Test convergence of $\sum_{n=1}^{\infty} \dfrac {1}{\log(e^n+e^{-n})}$ Attempt: I have tried the integral test, the comparison test ( for which I couldn't find a suitable comparator). However, I ...
3
votes
3answers
84 views

Where does the sum of $\sin(n)$ formula come from?

I have seen Lagrange's formula for the sum of $\sin(n)$ from $1$ to $n$ during one of my classes last week, but I never saw how it came to be. I tried googling it to find a proof but couldn't seem to ...
2
votes
1answer
17 views

How prove $\sum_{i=2}^na_i^{1-\frac{1}{i}} < S+2\sqrt{S}$ for $S=a_2+\dots +a_n.$

Let $a_2,\dots,a_n>0$ and $S=a_2+\dots +a_n.$ How prove $\sum_{i=2}^na_i^{1-\frac{1}{i}} < S+2\sqrt{S}.$
0
votes
1answer
35 views

Find a Taylor series around $x=0$ [on hold]

I don't know how to find the Taylor series around $x=0$ for: $$f(x)=\frac{\tan(2x)-\arctan(4\sinh(x))}{\sin(x^{2})}$$ Thank you in advance.
0
votes
3answers
100 views

convergence of $(1+\frac{1}{2})(1+\frac{1}{4})\ldots(1+\frac{1}{2^n})$

I was given a problem of testing sequence convergence. The sequence is defined as: $$x_n= (1+\frac{1}{2})(1+\frac{1}{4})\ldots(1+\frac{1}{2^n})$$ My first idea was to define $y_n$ as follows: $$y_n = ...
2
votes
3answers
83 views

Simplify the sum $ \sum_{k=1}^{\infty} (\frac{1}{2})^kk $

I need some help simplifying this sum: $$ \sum_{k=1}^{\infty} \left(\frac{1}{2}\right)^kk $$ I have a feeling it's some basic series thing that I'm forgetting, but I need help nonetheless.
5
votes
2answers
103 views

What is the general term of $a_{n+1}=\frac{2a_n-1}{5a_n-1} \ , \ \ a_1=1$?

I've struggled to solve this exercise $$a_{n+1}=\frac{2a_n-1}{5a_n-1}\ , \ \ a_1=1$$ $$b_{n+1}=(5a_n-1)b_n \ , \ \ b_1=1$$ Find $b_{\ 40}$ . $$$$ I thought 'taking inverse' will be ...
1
vote
1answer
46 views

What kind of sequence is that ($1+2+2^2+\cdots+2^k$) and how it can be expressed in a short way?

I am curious what kind of sequence is that $$1+2+2^2+2^3 +\cdots+2^{k-1}$$ and how it can be simplified or expressed in some short way... In the classroom we expressed it as $2^{k-1}$ over something ...
2
votes
0answers
44 views

How can I calculate the sum of the following series? [duplicate]

I am trying to calculate the sum of S=(1-1/2 +1/3-1/4+...). I used wolfram Alpha but the answer makes no sense to me. Thank you in advance.
0
votes
0answers
16 views

Limit of sequence (limit of Bilateral sequence)

I have a question related sequence and limits of sequence. From definition we know that sequence is a function whose domain is natural number.Then we called a sequence (a_n) converges if for every ...
1
vote
0answers
25 views

Convergence of infinite products

I wonder, parallel to the theory of summability of infinite series is there a theory for infinite products? Is there any generalized convergence method (such as Cesaro and Abel summability) for the ...
2
votes
2answers
28 views

Measure of a modified Cantor set

Suppose a modified Cantor set: Starting with $E_0 = [0,1]$, we delete the middle interval of length $1/3$, then we delete the middle intervals of length $1/15$, and so on; in each step we delete from ...
1
vote
1answer
36 views

Find polynomial f(n) such that for all integers $n$ $\geq 1$, we have

Find polynomial f(n) such that for all integers $n \geq 1$, we have $3\left( 1\cdot2 + 2\cdot3 + \ldots + n(n+1) \right) = f(n)$. Write f(n) as a polynomial with terms in descending order of $n$.
4
votes
1answer
29 views

If $a_n$ and $b_n$ are equivalent sequences and $a_n$ is bounded then so is $b_n$.

This is what i know; If $(a_n)$ is an infinite sequence of which is bounded then we can say; $|a_i| < M $ for all $i \geq 0.$ since $a_n$ and $b_n$ are equivalent sequences, we can say that for ...
0
votes
2answers
27 views

Series with floor function - convergent?

I am trying to figure out where does the following series converge to as $n$ goes to infinity (if it doest at all) $$\frac{1}{n} \sum^{n}_{t=\lfloor \rho n \rfloor +1} \frac{n}{t}$$ where $\rho$ is ...
1
vote
2answers
58 views

Question about series convergence $\sum_{n=1}^\infty \frac{1}{n}$ and $\sum_{n=1}^\infty \frac{1}{n^2}$

So I have been playing around with convergent series recently and I still have a hard time understanding why $\sum_{n=1}^\infty \frac{1}{n}$ diverges and $\sum_{n=1}^\infty \frac{1}{n^2}$ converges. ...
0
votes
1answer
43 views

Calculate this infinite sum [duplicate]

$$s= \sum_{n=1}^\infty \frac{n+3}{(2^n)(n+1)(n+2)}$$ Any method to calculate this type of infinite sums?
0
votes
0answers
36 views

Is this approach to the Collatz conjecture flawed?

This question has 3 parts. 1/ outlines a "shadow function"; 2/ uses this shadow function to show there are no non-trivial loops; 3/ uses the notion of "combs" to show there are no sequences ...
1
vote
0answers
39 views

Closed form for the summation $\sum_{k=1}^n\dfrac{1}{r^{k^2}}$

Is there any closed form for the finite sum $$\sum_{k=1}^n\dfrac{1}{r^{k^2}}$$ or infinite sum ( when $|r|<1$) $$\sum_{k=1}^\infty\dfrac{1}{r^{k^2}} ?$$ While solving this problem, I found this ...
0
votes
3answers
39 views

Convergence of sequence s.t $|a_{n+1}-a_n|\le \frac{n^2}{2^n}$

Let $a_n $ be sequence of real numbers such that $|a_{n+1}-a_n|\le \frac{n^2}{2^n} $for all n $\in $ N. Then $ a_n $ is convergent $ a_n $ is bounded but not convergent $ a_n $ has 2 limit points. ...
0
votes
1answer
27 views

How to solve the recursion $f(n+2)=3f(n+1)-2f(n)+5$?

$$f(n+2)=3f(n+1)-2f(n)+5, \text{ with } f(1)=4, f(2)=5\\ f(n+2)=3f(n+1)-2f(n)+n, \text{ with } f(1)=4, f(2)=5$$ I can't find anywhere the solution for sequences of this type and am unable to figure ...
2
votes
1answer
54 views

closed form for a double sum

How can I prove that $$\underset{k\geq1}{\sum}\left(\underset{m=-\infty}{\overset{\infty}{\sum}}\frac{\left(-1\right)^{m}}{\left(2k-1\right)^{2}+m^{2}}\right)=\frac{\pi\log\left(2\right)}{8}\,?$$I ...
0
votes
1answer
30 views

If there is a positive constant $r$ such that $c_n \geq r >0 ~\forall~n\in \mathbb N$, then prove that $\sum_{n=1}^\infty a_n $ converges

Let $\{a_n\}, \{b_n\}$ be positive sequences. Let $c_n= b_n-\dfrac {b_{n+1}a_{n+1}} {a_n}$. Prove that : If there is a positive constant $r$ such that $c_n \geq r >0 ~\forall~n\in \mathbb N$, ...
-2
votes
1answer
22 views

Uniformly convergent & point convergent

Let ${({f_n})_n}$ is a sequence that ${f_n}(x)=tan^{-1}(nx), x\in [0,\infty)$. Prove for every $[a,b]$ that $a>0$ is Uniformly convergent and on $[0,b]$ just point wise convergent.
1
vote
2answers
39 views

Convergence of $\sum_{n=1}^\infty \frac {n^3[\sqrt 2 + (-1)^n]^n}{1.05^n} $

Test convergence of $\sum_{n=1}^\infty \dfrac {n^3[\sqrt 2 + (-1)^n]^n}{1.05^n} $ Attempt: By using the the $n^{th}$ root test : $\lim_{n \rightarrow \infty} a_n^{1/n} = \lim_{n \rightarrow \infty} ...
-1
votes
1answer
42 views

absolutely convergent & conditionally convergent [on hold]

Prove that $$\sum_{n=1}^{\infty}\left(\sum_{m=1}^{n}\frac{{{1}}}{{{m}}}\right)\frac{{{\sin(nx)}}}{{{n}}} $$ for $x = {k\pi}$ , $k\in \mathbb{Z} $ is absolutely convergent. & for $x ...
-4
votes
0answers
78 views

Explain about absolute convergence and convergence [on hold]

$$\sum_{n=1}^{\infty}\frac{{{(-1)^{n}\sin (n)}}}{{{n}}} $$ Explain about absolutely convergent and convergent. Thx a lot.
1
vote
0answers
19 views

how to prove $\Phi(t)$ is divergent when $Im(t)=\pi/2$?

The Riemann $\Xi(z)$ function admits a Fourier transform of a even kernel $\Phi(t)=3e^{5t/4}\theta'(e^{t})+2e^{9t/4}\theta''(e^{t})$. Here $\theta(z)$ is the Jacobi theta function. ...
0
votes
2answers
57 views

Prove$\sum_{n=0}^{\infty} \frac{n}{(n+1)^2}-\frac{1}{n+2} $ is convergent [on hold]

prove $$\sum_{n=0}^{\infty}\frac{{{n}}}{{{(n+1)^{2}}}}-\frac{{{1}}}{{{(n+2)}}} $$ is convergent.
0
votes
2answers
42 views

$ \sum_{n=1}^{\infty} \frac{{{n+1}}}{{{n}}}{a_n} $ is absolutely convergent. [on hold]

prove that if $ \sum_{n=1}^{\infty} {a_n} $ is absolutely convergent $ \sum_{n=1}^{\infty} \frac{{{n+1}}}{{{n}}}{a_n} $ is absolutely convergent.
0
votes
0answers
22 views

Prove the convergence and find limit [duplicate]

Let $x_n$ be a sequence with general formula: $$x_{n+1}=\sqrt{2+x_n}$$ I am supposed to prove it's convergence and find the limit as $n\to +\infty$ I thought of proving this sequence to be monotone ...
0
votes
1answer
22 views

prove that Radius of convergence is 1 [on hold]

Let's assume that $${\left\{ {{a_n}} \right\}_{n \in {\Bbb N}}}$$ is a positive sequence number. and let $$ \mathop {\lim }\limits_{k \to \infty }{A_k}=\sum_{n=0}^{k} {a_n} = \infty $$ if $$ \mathop ...
0
votes
1answer
22 views

Divisibility of binomial coefficients

I have got this series of binomial coefficients - $${2n\choose 0}+3\times{2n\choose 2}+3^{2}\times{2n\choose 4}+\ldots +3^{n}\times{2n\choose 2n}$$ I have to prove this to be divisble by ...
1
vote
1answer
35 views

Showing that the complete elliptic integral of the second kind can be represented by a particular series

I'm trying to show that the complete elliptic integral of the second kind can be represented as: $$E(k)=\frac{\pi}2 ...
0
votes
1answer
29 views

Convergence of $\sum_{n=1}^\infty{\frac{\ln({3n^2 +4n+5})}{n^{4/3}}}$

How can I test convergence for this series? I used limit comparison test with $\frac{1}{n^{5/4}}$ and seems to work, however I am looking for a simpler solution. $$\sum_{n=1}^\infty{\frac{\ln({3n^2 ...
5
votes
3answers
128 views

A numerical evaluation of $\sum_{n=1}^{\infty}(-1)^{\frac{n(n+1)}{2}}\frac1{n!}\int_0^1(x)_n dx$

I would like to obtain a numerical evaluation of the series $$S=\sum_{n=1}^{\infty}(-1)^{\frac{n(n+1)}{2}}\frac1{n!}\int_0^1x(x+1)\cdots(x+n-1) dx$$ to five significant digits. I've used Mathematica, ...
1
vote
2answers
48 views

Find the sum of the series $\sum \limits_{n=1}^{\infty} \frac{2^{-n}(n+3)}{(n+1)(n+2)}$ [on hold]

Find the sum of $\sum \limits_{n=1}^{\infty} \frac{2^{-n}(n+3)}{(n+1)(n+2)}$
1
vote
1answer
58 views

Is this equality true? Why? Why not?

Let $$ \lim_{a\to 0} \frac{1}{2} \left( \left( \sum_{n=-\infty}^\infty \frac{1}{(n+a)^2} - \frac{1}{a^2} \right) \right) = \sum_{n=1}^\infty \frac{1}{n^2}$$ I already know that ...