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

learn more… | top users | synonyms (5)

1
vote
1answer
20 views

Is the product of two monotone sequences monotone?

Question: The product of monotone sequences is monotone, T or F? Uncompleted Solution: There are four cases from considering each of two monotone sequences, increasing or decreasing. CASE I: Suppose ...
3
votes
1answer
35 views

Is $\lim na_n \to 0$ sufficient for $\sum a_n$ to converge?

Is $\lim na_n \to 0$ sufficient for $\sum a_n$ to converge? Or additional criteria is required? E.g. $a_n$ needs to be positive? Is naïve comparison with $\frac {1}{n^p}$ series justifies that ? Or is ...
0
votes
1answer
13 views

Find the ratio and interval of convergence for $\sum_{n=1}^{\infty} \frac{n!x^n}{1\cdot 3\cdot 5\cdots (2n-1)}$

I believe this would diverge for $x\neq 0$. After using the ratio test I obtain (x)(n+1)(sum from 1 to n of (2n-1)/(2n+1)). Taking the limit as n goes to infinity the second term blows up and the ...
2
votes
1answer
44 views

Prove the sequence determined by $a_{n+1}={a_n\over \sin a_n}$ is convergent, and found its limit.

Let $\{a_n\}$ be a sequence defined by $0<a_1<{\pi \over 2}$, $a_{n+1}={a_n\over \sin a_n}$. $Attempt:$ $a_1>0$ and $\sin a_1>0$ and therefore the sequence begins positive and remains ...
4
votes
1answer
49 views

Evaluating $\lim_{n \to \infty} \sum\limits_{k=1}^n \frac{1}{k 2^k} $

Question: How to compute $$ \lim_{n \to \infty} \sum\limits_{k=1}^n \frac{1}{k 2^k}? $$ Here is what I have tried so far: Define $s_n=\sum\limits_{k=1}^n \frac{1}{k 2^k}$ for every index $n$, ...
-2
votes
2answers
13 views

Find closed form for a sequence using the Fibonacci and Lucas number sequences. [on hold]

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

Easy Analysis question [on hold]

Prove $\{\sqrt{n+1}-\sqrt{n}\}$, $n ≥ 0$, is monotone, using just algebra
2
votes
3answers
26 views

Explicit (or recursive) formula of a sequence

Is there an explicit or recursive formula for this sequence starting from n=1: 1, -1 , -1 , 1 , 1 , -1 , -1 , 1 , 1 , -1 , -1 , 1 , 1 , -1 , -1 , 1 , 1 , ...
0
votes
2answers
36 views

proving a limit by definition

given a sequence $a_n=a^{\frac{1}{n}}$ for $n\in\mathbb{N}^*$, $a\in\mathbb{R},a>1$ then proof that $\lim\limits_{n\to+\infty}a_n=1$ by definition. proof: given $a_n=a^{\frac{1}{n}}$ for ...
0
votes
1answer
32 views

Convergent series proof!

Let $S$ be a non-empty subset of $\Bbb{R}$ that is bounded above. Show that there exists a sequence $(a_n)_{n\in \Bbb{N}}$ contained in $S$ (that is, $a_n \in S$ for all $n \in \Bbb{N}$) which is ...
0
votes
2answers
22 views

Determine if the series diverge or converge?

So I was wondering what is the best TEST(divergent test, alternating series,power series,ratio test or root test) that we can use for the following series :
1
vote
1answer
20 views

Infinite Sum with differential operator

How would one suggest to calculate the following sum? $\sum^{∞}_{n=1}\partial_{x}^{2n}(\frac{\pi}{2x}Erf[\frac{cx}{2}])=?$ where c is just a constant. cheers.
-1
votes
0answers
10 views

need help solving this series [on hold]

i'm finding it difficult finding if this series converges or diverges. any help is appreciated. $\sum _{n=0}^{\infty }\left(3^{2+n}2^{1-3n}\right)$
0
votes
1answer
7 views

Problem with finding generating function for a sequence

Problem: Determine the generating function for the sequence $(a_k)$ given by $a_0 = 2$ and $a_k = 3a_{k-1} - 4$ for $k \geq 1$. Solution: We define $f(x) = \sum_{k=0}^{\infty} a_k x^k$ for the ...
1
vote
1answer
15 views

Prove that for every $T > \frac{\pi}{2} $, $\int_{\frac{\pi}{2}}^T \frac{cos(x)}{x}dx < 0$

I tried doing integration by parts a few times, after doing it 3 times I get the following expression: $$ \int_{\frac{\pi}{2}}^T \frac{cos(x)}{x}dx = \frac{sin(T)}{T} - \frac{1}{\frac{\pi}{2}} - ...
1
vote
1answer
22 views

Why $\frac{1}{n}\sum_{j=1}^mj^p \asymp\frac{1}{n}m^{p+1}$ as $n\to\infty$?

Why $$\frac{1}{n}\sum_{j=1}^mj^p \asymp\frac{1}{n}m^{p+1}$$ as $n\to\infty$, where $p>0$ and $a_n\asymp b_n$ if and ony if there exists a constant $c^{-1} \leq b_n/a_n \leq c$?
1
vote
0answers
41 views

Series of form $\sum_{n=1}^{\infty}f_n(x)$

I have the following series $$\sum_{n=1}^{\infty}f_n(x)$$ where $f_n(x)=\frac{(-1)^n}{n^2}\cos(n\pi x)$. I have to prove that the series converges absolutely and uniformly on the interval $[-1,1]$. ...
4
votes
2answers
41 views

$\lim_{n \rightarrow \infty} \frac {1^{a+1}+2^{a+1}+\cdots+n^{a+1}}{n.(1^{a }+2^{a }+\cdots+n^{a })} $

The value of $$\lim_{n=\infty} \dfrac {1^{a+1}+2^{a+1}+\cdots+n^{a+1}}{n.(1^{a }+2^{a }+\cdots+n^{a })} $$ Attempt: $S = \lim_{n \rightarrow \infty} \sum_{n=0} ^\infty \dfrac {k^{a+1}} {n.( 1^{a ...
5
votes
1answer
41 views

How do I solve this infinitely nested radical? [duplicate]

$\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+...}}}}$ Apparently, the answer is 3.
-3
votes
0answers
31 views

Why $\zeta(-1)=-1/12$? Doesn't defining it like this instead create problems? [duplicate]

Why $\zeta(-1)=-1/12$? As $$\zeta(-1)=\sum_{k=1}^{\infty}k\sim\infty$$How does it makes sense? Also about other $\zeta(t)\mid t<0$
1
vote
1answer
36 views

Can a divergent alternating series by rearrangement of terms be made to converge to a value?

Riemann discovered that a conditionally convergent series, through rearrangement of it's terms, can be made to converge to any value. But, if $S$ is a divergent alternating series, through ...
2
votes
1answer
28 views

How to calculate the limit of this sequence which incorporates tan?

I was revising for my pre-calculus exam, which is in two weeks time, and I started proving some sequence related theorems. I got interested in limits and I started deepening the concept. I got to a ...
0
votes
0answers
19 views

Is it possible to show the uniqueness of formula for solution?

The motivation to this question can be found in: Show that any sequence $(u_{n})$ must tends to infinity as $n→∞$ My question is: Is it possible to show the uniqueness of the formula for the ...
1
vote
0answers
16 views

Problem with coefficient of $x$ in product expansion of $\sin(x)$ (Basel Problem)

I'm working on the Basel problem, and in my working I have $\sin(x)=Ax(x-\pi)(x+\pi)(x-2\pi)(x+2\pi)...$ But I think I have a problem with the coefficient of x - wouldn't it be infinitesimal, if we ...
1
vote
1answer
35 views

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

Convergence of $$\sum_{n=1}^\infty n^s(\sqrt {n+1} - 2 \sqrt n + \sqrt {n-1})$$ Attempt: $\sum_{n=1}^\infty n^s(\sqrt {n+1} - 2 \sqrt n + \sqrt {n-1}) \sim \sum_{n=1}^\infty n^s( \sqrt n )$ As ...
3
votes
1answer
68 views

Prove 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 $\implies \sum \left(\frac {b_n}{a_n}\right)$ ...
0
votes
1answer
32 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
32 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
31 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
1answer
21 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
10 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
17 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
26 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
20 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
39 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
1answer
68 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
36 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
46 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
89 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
53 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
79 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
62 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
38 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
88 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
64 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
8 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
112 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} ...