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

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31
votes
5answers
791 views

About a harmonic series problem : How to prove that $\sum_{n=1}^{\infty}\frac{H_n}{n^3}=\frac{\pi^4}{72}$

How to prove that $$\sum_{n=1}^{\infty}\frac{H_n}{n^3}=\frac{\pi^4}{72}$$ $H_n$ is the n th harmonic number
1
vote
1answer
52 views

Prove that if $x_n\to x$ and $x > r$, then there exists $K\in\Bbb N$ such that $n\geq K\implies x_n > r$.

Let $(x_n)_{n\in \mathbb{N}}$ be a convergent sequence with limit $x$ and $x>r$. Is there a $K \in \mathbb{N}$ such that $n \geq K \implies x_n > r.$
2
votes
2answers
170 views

If $\sum_{n}A_n$ is convergent, then $\sum_{n} (-1)^{n} A_n$ is also convergent?

If you know that the series $\sum_{n}A_n$ is convergent, i have to prove or give a counterexample of the following statement: The series $\sum_{n} (-1)^{n} A_n$ is also convergent. Please help with ...
0
votes
2answers
83 views

Calculate the limit of a sequence

A sequence $c_n$ is defined by the following recursion $c_{n+1} = c_n + c_{n-1}$ for every $n \geq 1$ and $c_0 = 1, c_1 = 2$. -Let $a_n = \frac{c_{n+1}}{c_n}$, for every $n\geq 0$ and prove that $a_n ...
3
votes
2answers
61 views

Evaluate $\displaystyle\mathop {\lim }\limits_{n \to \infty } {\left( {\frac{{{a^n} + {a^{2n}}}}{{1 + a}}} \right)^{1/n}}$ where $0<a<1$

Here is my working, feel free to add comments if you see anything wrong with it, thanks! Since $0<a<1$, we have $a^{n}<a^{n}+a^{2n}<2a^{n}$, hence $a{\left( {\frac{1}{{1 + a}}} ...
2
votes
3answers
267 views
5
votes
3answers
103 views

Stuck on this integral involving exp and the floor function

Here is the integral $$\int_0^\infty \lfloor x \rfloor e^{-x}dx$$ Here is what I have so far: $$I = \sum_{n=0}^\infty \int_n^{n+1} n e^{-x}dx$$ $$ = \sum_{n=0}^\infty -ne^{-n-1} + ne^{-n}$$ $$ = ...
3
votes
2answers
53 views

Is this limit correct?

In $\mathbb R$, Suppose $\{x_n\}\to 0 $ and $\{y_n\}$ is bounded. In order to prove $\lim_{n\to \infty}x_ny_n=0$, I proved: $\lim_{n\to \infty}|x_ny_n|\leq M\lim_{n\to\infty}x=0$, and conclude the ...
0
votes
0answers
207 views

Show a series of functions has a continuous sum.

For every $x \in \mathbb{R}$ define $$I(x) = \begin{cases} 0 & \text{if} & x \leq 0,\\ 1 & \text{if} & x > 0 \end{cases}$$ Suppose that $(x_n)$ is a sequence of points in $(a, b)$ ...
1
vote
2answers
293 views

Determine the exact interval of convergence of $\sum x^{n!}$

Determine the exact interval of convergence of the following power series: $$\sum x^{n!}$$ Theorem 23.1 For the power series $\sum a_nx^n$, let $\beta=\limsup |a_n|^{1/n}$ and $R=1/\beta$. The power ...
2
votes
0answers
104 views

Two trigonometric and exponential integral

You may view my related problem: A hard definite integral with trignometric Show that : $$\int_0^{\frac{\pi}{2}}x\sqrt{\tan x}\text{e}^{2ix}\text{d}x=\frac{\pi ...
5
votes
1answer
142 views

Sequence involving primes of form $n^2 + n+1$

Looking at prime numbers $p_i $ of the form $n^2+n+1$ and the derived expression $$1 - \prod_{i=1}^{j}\frac{(p_i-1)}{p_i}$$ it seems (I do not claim it and do not see why it should be true) that ...
22
votes
1answer
605 views

Evaluate $\sum\limits_{k=1}^{\infty} \frac{k^2-1}{k^4+k^2+1}$

How to find $$\sum_{k=1}^{\infty} \frac{k^2-1}{k^4+k^2+1}$$ I try something like this: $$\begin{align*}\sum_{k=1}^{\infty} ...
8
votes
2answers
362 views

A tough series: $\sum_{k=1}^{\infty}\frac{\zeta(2k)}{(k+1)(2k+1)}$, need help

I was doing a integral which ends up with a tough series part: $$\sum_{k=1}^{\infty}\frac{\zeta(2k)}{(k+1)(2k+1)}$$ Mathematica says $$\frac12$$ Which agrees with the anwer...Anyone know how to ...
0
votes
1answer
102 views

Convergence of Sequences of Functions

Suppose that $E$ is a countable set. For every $n \geq 1$, consider a function $f_n: E \to \mathbb{R}$ in such a way that the sequence $(f_n)$ is pointwise bounded. Show that there is a subsequence ...
12
votes
3answers
774 views

Summation of infinite series with hyperbolic sine

The following is a conjecture. I would like to prove that $$\sum_{n=0}^\infty \frac{1}{(2n+1)\operatorname{sinh}((2n+1)\pi)}=\frac{\log(2)}{8}.$$ Both sides agree to at least $100$ digits, so I ...
4
votes
2answers
214 views

If the sum converges to zero, does that mean that each sequence converges to zero?

If we are given that the sequence sum of two sequences of positive real numbers converges to zero, does that mean that each sequence converges to zero? (by the squeeze theorem)
3
votes
1answer
117 views

Problem showing a certain series converges

I'm trying to show the series ${\displaystyle \sum_{n=1}^{\infty}\frac{1}{\sqrt{n}}\sin\left(nx\right)}$ converges for all $x\in\left[0,2\pi\right]$. Using Dirchlet's Test it suffices to show that ...
6
votes
3answers
185 views

Infinite series : $\sum_{n=1}^{\infty}\prod_{i=1}^n\frac{(3i-1)}{(4i-3)} $

How to evaluate this? $$\sum_{n=1}^{\infty}\prod_{i=1}^n\frac{(3i-1)}{(4i-3)} $$
1
vote
3answers
56 views

Need help with series definition

This is not a homework problem. I really tried to solve it on my own for some time but haven't gotten far. I suspect this problem may exceed my fairly rusty high-school math. Or, the solution is ...
1
vote
1answer
115 views

Techniques to prove properties of a sequence

What techniques/methods can be used to prove that the sequence produced by $n\cdot (n+1)\cdot (2\cdot n+1)/6$ contains only one square ($4900$) greater than 1? While this particular sequence is an ...
1
vote
1answer
478 views

Is $\sum \sin^2(k)/k$ Convergent? [duplicate]

A student recently used the series $\displaystyle\sum_{k=1}^\infty\frac{\sin^2k}{k}$ as an example of a divergent series whose terms tend to $0$. However, I'm having trouble convincing myself that ...
1
vote
0answers
65 views

what are the borders of the convergence disks of series?

Let $\mathbb{T}=\{z\in \mathbb{C}\mid |z|=1\}$. For which $S\subseteq \mathbb{T}$, is there a sequence $(a_n)\subseteq \mathbb{C}$ such that the series: $$\sum_{k=1}^\infty{a_kz^k}$$ is convergent on ...
2
votes
2answers
3k views

Monotonically increasing vs Non-decreasing [duplicate]

Is monotonically increasing is same as non-decreasing? Thank you for answer beforehand.
0
votes
1answer
55 views

Calculate $\sum_{n=1}^{N}z^n$ and show that if $z\not=1$ $|\sum_{n=1}^Nz^n |\leq \frac{2}{|1-z|}$ and $\sum_{n=1}^\infty \frac{z^n}{n}$ converges.

Let $z\in\mathbb{C}$ and $|z|=1$. Calculate $\sum_{n=1}^{N}z^n$ and show that if $z\not=1$ $$|\sum_{n=1}^Nz^n |\leq \frac{2}{|1-z|}$$ and $\sum_{n=1}^\infty \frac{z^n}{n}$ converges. This is what I ...
5
votes
3answers
257 views

$\sum a_n$ converges $\iff \sum (\sqrt{1+a_n}-1)$ converges

Let $a_n\in\mathbb{R}_{\geq 0}$ and assume $\sum {a_n}^2$ converges. Show that: $\sum a_n$ converges $\iff \sum (\sqrt{1+a_n}-1)$ converges For $\Rightarrow$ I think I need to show that ...
4
votes
2answers
255 views

Is there a closed form expression for the first half of the Binomial series?

I'm looking for a closed form expression for the sum $P_n(x) =\sum_{0\leq k\leq n/2}\binom{n}{k}x^k$, where $n$ is a given positive integer and $k$ runs over nonnegative integers between $0$ and ...
0
votes
1answer
99 views

Maclaurin vs Taylor and their geometrical difference

In this topic i learned how to approximate a function with a high degree polynomial and how to derive the Maclaurin series: $$ f (x) = P_n(x) = f(0)+{f'(0)\over 1!}x+{f''(0)\over ...
2
votes
2answers
175 views

A integral with polygamma

I was doing a integral, the last part is $$\int_0^{\frac{\pi}{2}}x^3\csc x\text{d}x$$ I ran this on Maple, it turns into polygammas...How we evaluate this? I think there should be a way to evaluate ...
8
votes
1answer
405 views

Summation of an Infinite Series: $\sum_{n=1}^\infty \frac{4^{2n}}{n^3 \binom{2n}{n}^2} = 8\pi G-14\zeta(3)$

I am having trouble proving that $$\sum_{n=1}^\infty \frac{4^{2n}}{n^3 \binom{2n}{n}^2} = 8\pi G-14\zeta(3)$$ I know that $$\frac{2x \ \arcsin(x)}{\sqrt{1-x^2}} = \sum_{n=1}^\infty ...
0
votes
2answers
72 views

Error propagation through recurrence relation

I want to see how the error propagates on a mapping that I have. I have proven that $$|f(x+\varepsilon)-f(x)|=\varepsilon(1+\varepsilon),$$ let $\varepsilon_n$ be the error after $n$ applications of ...
0
votes
1answer
78 views

Need help with a integral

I was evaluating $$ \int_{0}^{^\pi/_2}x\ln\left(\vphantom{\large A}\cos\left(x\right)\right)\,{\rm d}x $$ I like to try with the fourier series $$ \int_{0}^{^\pi/_2} \left[\,\,\sum_{k = 1}^{\infty} ...
0
votes
1answer
102 views

Find the sums of the series?

$$ 1- \frac{1}{5 \cdot 3^2} - \frac{1}{7 \cdot 3^3} + \frac{1}{11 \cdot 3^5} + \frac{1}{13 \cdot 3^6} - - + + \cdots $$ Not really sure how to start the problem, and help will be grateful
2
votes
1answer
44 views

summation of series of powers $ n^{nix} $

is the series .. $$ \sum _{n=2}^{\infty}n^{kin} $$ here k is a real number is convergen or divergent ??, for example perhaps we can copare it to the series $$ \sum _{n=2}^{\infty}n^{ix} $$ which is ...
1
vote
1answer
71 views

$a_{n}b_{n} \rightarrow 0$ if $(a_{n})$ is bounded and $(b_n)$ converges to $0$

I have trouble proving the problem below. Any tips? Suppose that $(a_{n})$ is a bounded but not necessarily convergent sequence and that $(b_n)$ is a sequence converging to $0$. Prove that ...
0
votes
1answer
65 views

If $x_{n+1}=ax_{n}^b$, find $x_{n}$

Given $x_{1}$, $a$, $b$, and $x_{n+1} = a x_{n} ^ b$, find $x_{n}$. I'm not really sure how to tackle this problem. I do know that with all $a$ and $b$ that I will be using, the sequence will ...
2
votes
3answers
103 views

Does the series $\sum_{n=1}^\infty \frac{1}{n^p}$ converge for $p\in [1, +\infty)$

Does the series $$\sum_{n=1}^\infty \frac{1}{n^p}$$ converge for $p\in [1, +\infty)$? If it does not, how to prove it?
0
votes
1answer
90 views

A improper integral with series

Show that $$\begin{align} & \int_{-\infty }^{+\infty }\frac{\cos \alpha x}{( \beta^2+x^2 )( ( \beta +1 )^2+x^2 )\cdots ( ( \beta +n )^2+x^2 )} \, \text{d}x = 2\pi \sum\limits_{k=0}^n ( -1 )^k ...
0
votes
0answers
59 views

logistic difference equation problem

Consider logistic difference equation $${{x}_{n+1}}-r{{x}_{n}}\left( 1-{{x}_{n}} \right)=f\left( x \right),\ \ 0\le {{x}_{n}}\le 1\ \ \ \ \ \ \left( 1 \right)$$ 1.Show hat expression $$f\left( f\left( ...
2
votes
4answers
2k views

Relationship between limsup, liminf and limit.

If you want to show that a sequence $(a_{n})$ in $\mathbb{R}$ is convergent, when is it sufficient to show that there is a number $b\in\mathbb{R}$ such that $$ \liminf a_{n} \geq b \geq \limsup ...
1
vote
4answers
124 views

“Differs in only finitely many terms” an equivalence relation on sequences?

Consider sequences $a : \mathbb{Z}^+ \rightarrow A$ on a set $A$. Define the relation $\sim$ over sequences by $a \sim b$ iff there are only finitely many indices $i$ at which $a_i \neq b_i$. Clearly ...
6
votes
4answers
262 views

Construct a convergent series of positive terms with $\displaystyle\limsup_{n\to\infty} {a_{n+1}\over{a_n}}=\infty$

Construct a convergent series of positive terms with $\displaystyle\limsup_{n\to\infty} {a_{n+1}\over{a_n}}=\infty$ My thoughts: By the theorem: Suppose $a_n\ge0$ for all $n$, and let ...
3
votes
2answers
311 views

Show if $\displaystyle\sum\limits_{k=1}^\infty {a_k}^2$,$\displaystyle\sum\limits_{k=1}^\infty {b_k}^2$ converge, their product converges too

Show if $\displaystyle\sum\limits_{k=1}^\infty {a_k}^2$ and $\displaystyle\sum\limits_{k=1}^\infty {b_k}^2$ both converge, $\displaystyle\sum\limits_{k=1}^\infty {a_kb_k}$ also converge then Show if ...
0
votes
2answers
39 views

If two infinites series obey an inequality from $k = N,\ldots,\infty$ does this imply that the inequality is obeyed for $k=1,\ldots, \infty$?

If $$ \frac{1}{k^{\log(k)}} < \frac{1}{k^2} $$ for $100 < k$, does this automatically imply that: $$ \sum_{k=1}^\infty \frac{1}{k^{\log(k)}} < \sum_{k=1}^\infty \frac{1}{k^2}, $$ or only ...
0
votes
1answer
102 views

Simplification of double series

Consider a double series in the following form: $$\sum_{j,k=0}^\infty\frac{1}{a_jb_k+c_jd_k}.$$ Is there a way of evaluating it, if $\sum_{j=0}^\infty a_j^{-1}$, $\sum_{k=0}^\infty b_k^{-1}$, etc... ...
1
vote
1answer
119 views

analysis double series sums

Let $a_{ij}$ be entries in a matrix in $i$th row and $j$th column such that $a_{ij} = \left\{ \begin{array}{l l} 0 & \quad \text{if $i$ < $j$}\\ -1 & \quad \text{if $i$ = $j$}\\ ...
2
votes
1answer
74 views

Please Help me with this complicated convergent test!

The question is: test $$\sum_k ((e^{k^{1/4}}+11)^{0.5}-(e^{k^{1/4}}-11)^{0.5})^k$$ for convergence. I believe I have to use the root test to get rid of the exponent of $0.5$, but after that I am ...
-2
votes
1answer
121 views

Power series of a function with multiplication

I don't understand what happens during series multiplication and substitution. It doesn't seem to make sense. Shouldn't it just be $\large(x^{n})^{n}$?
1
vote
1answer
178 views

Is there a forumla to find out if n faces can be made into a 'regular polyhedron'?

I'm not too sure about the exact terminology since Wikipedia is throwing me all over the place. I'm looking for a formula to find out if for n faces a 'regular polyhedron' can exist. In case that's ...
0
votes
2answers
304 views

Power Series Ratio Test

Last time ive checked the ratio test was limit to infinity of a+1 / a However, Ive approached a good amount of question that uses the inverse of that formula such as the following given below. ...