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

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2
votes
1answer
45 views

Proe that If $a_n>0$ and $\sum a_n$ converges then $\sum (\frac {b_n}{a_n})$ converges

Let $\{a_n\},\{b_n\}$ be two sequences such that for each $n$ we have $e^{ a_n }= a_n + e^{b_n }$. Show that if $a_n>0$ and $\sum a_n$ converges $\implies \sum (\dfrac {b_n}{a_n})$ converges ...
0
votes
1answer
25 views

Limit of sequence $ S_n $=$\int_0^1 \frac{nx^{n-1}}{1+x}dx $ [duplicate]

Let $ S_n $=$\int_0^1 \frac{nx^{n-1}}{1+x}dx $ for n$ \ge $1 then as n tends to infinity sequence tends to: 1.0 2.1 3. 1/2 4. Infinity Is there any other way, than to first do integration, and then ...
0
votes
0answers
30 views

Is there a way to reverse the ratio test?

My question arises from the following problem: Let $ a_n $ be a real series, so that $ \sum_{n=1}^ \infty a_n $ converges and $a_n \ge 0 $ and $a_n$ monotonously decreasing. It is to prove: $ ...
2
votes
0answers
29 views

$\lim_{n \to \infty}\prod_{k=1}^n \cos(k\sqrt{\frac{3}{n^3}} t) = e^{- \frac{t^2}{2}}$ [duplicate]

Is it true that $$\lim_{n \to \infty}\prod_{k=1}^n \cos\left(k\sqrt{\frac{3}{n^3}} t\right) = e^{- \frac{t^2}{2}}$$ ? How to proceed ?
0
votes
0answers
15 views

Closed form solution of a summation

First off I have absolutely no clue what I'm doing, my notes given for this course do not explain anything and I'm not sure if I'm doing this properly so I'm looking for help and an explanation on how ...
0
votes
0answers
7 views

Convergence of a Sequence in l^1 space

If a,b are complex numbers, and $k\ \epsilon$ N, the sequence $x_k = a+b^k$ will belong to $l^1(N) = \lbrace (c_0,c_1... ) : \sum_{k=0}^{\infty} |c_k| < \infty \rbrace$ for which a,b? a and b ...
0
votes
1answer
15 views

Series increasing or decreasing with factorials

I have been working on some homework for calc 3 and my prof has put a couple sequences in which we must find if they are increasing or decreasing with factorials in them. I've googled and there are ...
1
vote
1answer
24 views

Convergence of $\sum_{k=1}^\infty k\sin\frac1k$

$\sum_{k=1}^\infty k\sin\frac1k$. Can anyone provide me with a hint? Which test would help me in this situation?
0
votes
1answer
14 views

How to calculate the sum of a general series

In class we learned how to test the convergence of series and how to calculate the sums of arithmetic and geometric series (if they exist) but are there methods to actually calculate the values ...
1
vote
2answers
36 views

A uniformly convergent sequence of real analytic functions which does not converge to a real analytic function

I am looking for an example of a uniformly convergent sequence of real analytic functions which does not converge to a real analytic function. Also I would appreciate any pointers on how to think of ...
1
vote
0answers
53 views

Determine the limit of a series, involving trigonometric functions: $\sum \frac{\sin(nx)}{n^3}$ and $\frac{\cos(nx)}{n^2}$

I have $$\sum^\infty_{n=1} \frac{\sin(nx)}{n^3}.$$ I did prove convergence: $0<\theta<1$ $$\left|\frac{\sin((n+1)x)n^3}{(n+1)^3\sin(nx)}\right|< \left|\frac{n^3}{(n+1)^3}\right|<\theta$$ ...
1
vote
1answer
19 views

Sequence forming a vector space

The sequences $(x_k)_{k=1}^{\infty}$ in $\mathbb R$ , all or almost all $\neq 0$ with operations defined component by component, form a vector space V over $\mathbb R$. Find a basis of V, ...
2
votes
3answers
35 views

If $\sum a_n$ converges absolutely , then so does, $\sum \frac {a_n^2} {1+a_n^2}$

If $\sum a_n$ converges absolutely , then so does, $\sum \dfrac {a_n^2} {1+a_n^2}$ Attempt: Given that $\sum a_n$ converges absolutely $\implies \sum |a_n|$ converges. ...
-1
votes
1answer
16 views

How do I find the convergence of this summation using the comparison test? (∑(1/√(n^3-n)))

How do I find the convergence of this summation using the comparison test? \begin{equation} \sum_{n=1}^{\infty} \frac{1}{\sqrt{n^3 - n}} \end{equation} I am not sure what the comparison sequence ...
0
votes
1answer
41 views

Prove that if $\sum |a_n|$ converges, then $\sum a_n^2$ also converges. [duplicate]

Prove that if $\sum |a_n|$ converges, then $\sum a_n^2$ also converges. Attempt: $\sum |a_n|$ converges $\implies \sum |a_n|<M$. If $\sum |a_n|$ converges, then $\sum a_n $ also converges. ...
5
votes
1answer
88 views

Study the convergence of $\sum_{n=1}^\infty \frac{(-1)^n \cos^2(n)}{n}$

Study the convergence of $\sum_{n=1}^\infty \frac{(-1)^n \cos^2(n)}{n}$ Abel's/Dirichlet's tests cannot be applied here. I guess it's something more tricky involving integration maybe (?)
3
votes
2answers
51 views

convergence of $ \sum_{n=1}^{\infty} (-1)^n \frac{2^n \sin ^{2n}x }{n } $

Find values of $x$ for which the following series converges $$ \sum_{n=1}^{\infty} (-1)^n \dfrac{2^n \sin ^{2n}x }{n } $$ Attempt: (a) Check for Absolute Convergence If we consider $ ...
1
vote
4answers
77 views

Convergence divergence of $\sum \frac{n^4}{e^{n^2}}$

Check the Convergence divergence of $\sum \frac{n^4}{e^{n^2}}$ I applied ratio test. But I am not feeling sure. Is ratio test ok here, or some other way is possible.
4
votes
1answer
56 views

Infinite sum of products of four Bessel functions

The discrete Schrödinger equation for two interacting electrons in 1D under an electric field reads $$ E\psi_{mn}=[(m+n)F+U\delta_{mn}]\psi_{mn}-\psi_{m+1,n}-\psi_{m-1,n} -\psi_{m,n+1}-\psi_{m,n-1}\ . ...
2
votes
1answer
37 views

Sum of exponential square series

I have a infinite sum which I wonder if it will converge to a simpler function $f(r) = \Sigma_n r^{n^2} , r<1$, I also interested in case $r$ is a complex number on unity circle $r = ...
2
votes
3answers
44 views

Radius of convergence of $\sum_{n=0}^\infty a_n z^{n^2}$

Find radius of convergence of power series $\sum_{n=0}^\infty a_n z^{n^2}$ where $a_0=1, a_n=3^{-n}a_{n-1}$ for n $ \in $N. I tried to get expression for $ a_n $ first which comes to be $ a_n$ ...
4
votes
1answer
86 views

show that $e=(\frac{2}{1})^{\frac{1}{1}}(\frac{4}{3})^{\frac{1}{2}}(\frac{6\cdot8}{5\cdot7})^{\frac{1}{4}}…$

I found $$e=(\frac{2}{1})^{\frac{1}{1}}(\frac{4}{3})^{\frac{1}{2}}(\frac{6\cdot8}{5\cdot7})^{\frac{1}{4}}(\frac{10\cdot12\cdot14\cdot16}{9\cdot11\cdot13\cdot15})^{\frac{1}{8}}.....$$ easily you can ...
1
vote
2answers
62 views

Sum of $1/n+1/(n-2) + 1/(n-4) + \cdots $

How does one calculate $$\frac{1}{n} + \frac{1}{n-2} + \frac{1}{n-4} \cdots $$ where this series continues until denominator is no longer positive? $n$ is some fixed constant positive integer.
0
votes
0answers
7 views

On Differentiation Theorem and Radius of Convergence

For reference: http://www0.maths.ox.ac.uk/system/files/coursematerial/2014/2644/48/14MT-AnalysisI-sheet7.pdf 4.(b) For fixed $d \in \mathbb{R}$, let $f_d(x)=S(d+x)C(d-x)+S(d-x)C(d+x)$ By ...
5
votes
3answers
103 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
27 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
29 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
67 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 ...
0
votes
1answer
50 views

Generating function $D(x) = (1 + x)(1+x^2)(1+x^3)\cdots$ [on hold]

Let $$D(x) = (1 + x)(1+x^2)(1+x^3)\cdots $$ 1) What is the inverse function of $D(x)$? 2) What sequence is generated by $D(x) $ Please don't vote down, the subject is complicated for me. Sorry ...
-2
votes
0answers
36 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
38 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 ...
-5
votes
1answer
24 views

Question of sequence [on hold]

Write the first and second differences of the sequence: 5,8,11,14,...
0
votes
1answer
21 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
59 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
43 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} ...
0
votes
2answers
77 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 …

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
65 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
90 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
21 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
103 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
84 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
105 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
47 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.
1
vote
1answer
21 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
34 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
29 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
40 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 $n$.