3
votes
2answers
47 views

Special values $\psi \left(\frac12\right)$ and $\psi \left(\frac13\right)$

I wonder if it is easy to prove that $$ \begin{align} \psi \left(\frac12\right) & = -\gamma - 2\ln 2, \\ \psi \left(\frac13\right) & = -\gamma + \frac\pi6\sqrt{3}- \frac32\ln 3, \end{align} ...
5
votes
3answers
267 views

Limit of iteration of sine function

Let $f(x) =\sin x $, and denote by $f^n = f\circ f\circ ...\circ f$ the n-th iteration of function $f.$ Find the limit (if exist) :$$\lim_{n\to\infty} n\cdot f^n (n^{-1} )$$
0
votes
1answer
27 views

If $|P_{n+1}-q|\le c|P_n - q|$ for all $n$, where $c<1$, then $P_n\to q$

Given $|P_{n+1}-q|\le c|P_n - q|$ for all $n$, where $c<1$ show that the $$\lim_{n\to\infty} P_n=q.$$ Was told to complete this problem by iteration. I'm terrible with proofs and they don't make ...
2
votes
1answer
32 views

Given a set of sequences, compute a corresponding set of functions

Consider the following set of sequences: $ S_k(n)= \begin{cases} 1 & \text{$n \equiv0\pmod{k}$}\\ 0 & \text{$n\not\equiv0\pmod{k}$}\\ \end{cases} $ I want to compute a set of ...
0
votes
0answers
28 views

Multiplying double sums formula

Questions: 1)Can I modify the same-endpoint formula somehow to get different-endpoint formula? http://en.wikipedia.org/wiki/Cauchy_product#Finite_summations 2)Or even better do you know any formula ...
-4
votes
1answer
65 views

Prove $\displaystyle\sum_{k=0}^\infty \frac{2^k}{\binom{2k+1}{k}}=\frac{\pi}{2}$ [on hold]

How to show that $$\sum_{k=0}^\infty \frac{2^k}{\binom{2k+1}{k}}=\frac{\pi}{2}$$
0
votes
3answers
52 views

finding a confusing limit

yeah just another limit. I have $Xn=\dfrac{1000^n}{\sqrt{n!}} + 1$ that $+1$ confuses me any hints & solutions how to calculate limit will be apreciated
1
vote
1answer
31 views

Which of the following series will converge and which one will diverge?

Can anyone help me out that which of the following series will converge and which one will diverge, with some explanation? A) $\sum_{n=1}^\infty \sin\left(\frac{\pi}n\right)$ B) $\sum_{n=1}^\infty ...
1
vote
1answer
43 views

Infinite Product implies divergence or not?

If $\displaystyle\prod_{n=1}^{\infty} (1-a_{n}) = 0$ then is it always true that $\displaystyle\sum_{n=1}^{\infty} a_{n} $ diverges? ($0 \leq a_{n} < 1) $
5
votes
1answer
125 views

Computing $\int_0^{\large1/1^2} \left\{ \frac{1}{t} \right\}\,\mathrm{d}t+\int_0^{\large 1/2^2} \left\{ \frac{1}{t} \right\}\,\mathrm{d}t+\cdots $

What tools would you recommend me for computing this series? $$\int_0^{\large1/1^2} \left\{ \frac{1}{t} \right\}\,\mathrm{d}t+\int_0^{\large 1/2^2} \left\{ \frac{1}{t} ...
2
votes
1answer
27 views

$\sum_{n=1}^{\infty} {(1-\cos(\sin 1/n))}^{w}$ with $w$ as parameter

Let $f(x)=(1-\cos(\sin x))$; $a_n=f(1/n)$ for $n\in\mathbb{N}$ For which $w>0$ series $$\sum_{n=1}^{\infty} {a_n}^{w}$$ converge? I haven't got a slicest idea how to check that, absolutely none ...
2
votes
1answer
43 views

Proving $\sum_{k=1}^{\infty}\frac{\sin kx}{x}=\frac{\pi-x}{2}$ for $0\le x\le 2\pi$

Refer to this OP: Sign of a series, we have the following equation \begin{equation} \sum_{k=1}^{\infty}\frac{\sin kx}{k}=\frac{\pi-x}{2} \end{equation} defined for $0\le x\le 2\pi$. Here is ...
-1
votes
0answers
43 views

How to find the result of infinite series $\sum\limits_{n=1}^\infty\frac{(-1)^{n+1}}{n^2}$? [duplicate]

As per wolfram-alpha the result is $\frac{\pi^2}{12}$. How to calculate it manually? $$\sum\limits_{n=1}^\infty\frac{(-1)^{n+1}}{n^2}=\frac{\pi^2}{12}$$
1
vote
1answer
21 views

How can I construct a specific sigmoid function?

The simple sigmoid function $$f(x)=1/(1+e^{−x})$$ approaches zero as x tends to negative infinity, and approaches $1$ as x tends to positive infinity. But I want to set $1$ and $20$ instead of $0$ and ...
-3
votes
0answers
21 views

Is the series convergent or divergent [closed]

If $\sum a_n$ with $a_n>0$ is convergent, then is $\sum \sqrt{a_na_{n+1}}$ always convergent?Either prove it or give a counterexample.
7
votes
0answers
96 views

Another way of expressing $\sum_{k=0}^{n} (-1)^k\frac{H_{k+1}}{n-k+1}$

In this post Another way of expressing $\sum_{k=0}^{n} \frac{H_{k+1}}{n-k+1}$ I asked for a solution of the non-alternating series. How about the alternating series? Can we find a nice way of ...
0
votes
0answers
22 views

A sequence of intervals- Trying to find a fixed point -

This might be a trivial question but I couldn't come up with a clever trick,theorems or whatnot. Suppose $I_0=\left[\frac{1}{h_0},\frac{1}{l_0}\right]$ where $h_0=1$ and $l_0=\frac{1}{2}$. Given ...
1
vote
2answers
30 views

Exist a function that satisfies these conditions?

Exist a function that safisties...? $$f(n)=\sum_{k=1}^{n}g(k)=n\ : g(k)\in (0,+\infty),\ g'(k)> 0, n\in \mathbb{N}$$ And this...? $$f(x)=\int_{0}^{x}g(t)dt=x\ : g(t)\in (0,+\infty),\ g'(t)> ...
1
vote
1answer
57 views

Summation of: $\sum_{1}^{\infty}\left(\frac{2}{3}\right)^x$ [duplicate]

This is a subsection in my statistics homework. It goes back to calculus II and summations, and it's been a long time since I've studied it so I'm rusty. I'm looking to solve the summation of ...
2
votes
7answers
137 views

Does the sequence converge, and to what? [closed]

We have a sequence $\{a_n\}$ $$a_0 = 0$$ $$a_{n+1} = \frac{a_{n}}{2} + 1$$ Does it converge? And to what?
0
votes
1answer
44 views

Is this series divergent or convergent?

I've been stuck with this problem for a couple of days trying to solve it but got no where till now. The problem states that we have to prove if the series given below is convergent or divergent, if ...
-2
votes
2answers
120 views

Convergency of $1+\frac{1}{4}-\frac{1}{9}-\frac{1}{16}+\frac{1}{25}+\frac{1}{36}-\frac{1}{49}-\frac{1}{64}+\cdots$ [closed]

Is this series convergent $1+\dfrac{1}{4}-\dfrac{1}{9}-\dfrac{1}{16}+\dfrac{1}{25}+\dfrac{1}{36}-\dfrac{1}{49}-\dfrac{1}{64}+\cdots\ ?$ Can we write this series as function of $n?$
2
votes
1answer
42 views

Show that certain sequence used in the proof of Wallis product formula is decreasing

Define $I_0 := \pi/2, I_1 := 1$ and $$ I_{n+2} := \frac{n+1}{n+2} I_n. $$ This sequence is monotone decreasing, which could be seen by recognizing that $$ I_n = \int_0^{\pi/2} \cos^n(x) \mathrm d x ...
4
votes
1answer
71 views

Test the convergence of the series $\sum \sin [\pi(\sqrt5+2)^n]$

I have just approached the following series $$\sum_{n=1}^\infty \sin [\pi(\sqrt5+2)^n]$$ And I already have a question. The $\lim_{n \to \infty}\pi(\sqrt5+2)^n=+\infty$. And the $\lim_{n\to \infty} ...
1
vote
1answer
28 views

Series radius of convergence

We have to find the values of $x$ for which the given series $\sum_{n=0}^{\infty}(-4)^n(x-5)^n$ is convergent. We know that a geometric series converges if $|r| < 1$. We start by expanding: ...
11
votes
5answers
175 views

Another way of expressing $\sum_{k=0}^{n} \frac{H_{k+1}}{n-k+1}$

Do you know any nice way of expressing $$\sum_{k=0}^{n} \frac{H_{k+1}}{n-k+1}$$ ? Some simple manipulations involving the integrals lead to an expression that also uses the hypergeometric series. ...
0
votes
1answer
31 views

how the ratio of two functions change…what am I doing wrong?

For $s \in \{1,\dots,T-2\}$, let $g(s) := \frac{f(s+1)}{f(s+2)} = \frac{\sum_{t=s+1}^{T} \frac{0.99^{t-1}}{1 + \text{exp}\left(\frac{t-1}{3} - 9\right)} \frac{1}{t}}{\sum_{t=s+2}^{T} ...
0
votes
1answer
17 views

Number of non-zero co-efficients in a series [closed]

Suppose $c_n \geq 0$ for all $n$ and $\sum_{n=0}^\infty c_nr^n$ converges for all $r\geq 1$, is it true that only finitely many $c_n$ are non-zero?
-2
votes
6answers
88 views

How to decide whether following series is convergent: $ \sum_{n=1}^{\infty} n^2\left( 2/3\right)^n$

How to decide whether the following series is convergent or otherwise: $ \sum_{n=1}^{\infty} n^2\left( 2/3\right)^n$? One way to do is to use root test, I am wondering if other ways are possible.
1
vote
1answer
35 views

Real Analysis Integral convex func [closed]

Given $f:\mathbb{R} \rightarrow \mathbb{R}$ with $\lambda \in (0,1)$ such that $$f(\lambda x+(1-\lambda)y) \leq \lambda f(x)+(1-\lambda)f(y) $$. Prove that $$\int_{0}^{2\pi} f(x) \cos{x} \mathrm{d}x ...
1
vote
5answers
89 views

Decide whether the following series converges $\sum_{n=1}^{\infty}\dfrac{(\ln n)^2}{n^{3/2}}$

Looking for a neat and smart way to solve this. I am having a tough time with this
1
vote
1answer
43 views

Does $f_n(a_n)\to f(a)$ hold?

Say, we have $f, f_n \in C^0(\mathbb R, \mathbb C)$ such that $f_n \xrightarrow{\text{uniform}}f$ and a sequence of reals $a_n \to a$. Does it then hold that $f_n(a_n)\to f(a)$? I couldn't think of ...
2
votes
2answers
41 views

formula for infinite sum of a geometric series with increasing term

I'm looking for the Expectation of the discrete random variable X, E[X], with pmf: $$p(x)=(\frac 16)^{x+1}, x=0,1,2,3...$$ so what I tried is as follows... $$E[X]= \sum_{0}^\infty xp(x) =$$ so then ...
4
votes
0answers
50 views

Simpler closed form for $\sum_{n=1}^\infty\frac{\Gamma\left(n+\frac{1}{2}\right)}{(2n+1)^4\,4^n\,n!}$

I'm trying to find a closed form of this sum: $$S=\sum_{n=1}^\infty\frac{\Gamma\left(n+\frac{1}{2}\right)}{(2n+1)^4\,4^n\,n!}.\tag{1}$$ WolframAlpha gives a large expressions containing multiple ...
0
votes
1answer
23 views

O Notation and taylor series

Wolframalpha tells me that the Taylor series of the exponential function is $1 + x + \frac{x^2}{2}+ O(x^3).$ Taylor series I just don't get this big O there, shouldn't this be a small o?
4
votes
3answers
58 views

Evaluate $ \lim n \left[ 1-\frac{(n+1)^n}{en^n}\right] $

Evaluate the following limit of sequence $$\lim_{n\to +\infty} n \left[ 1-\frac{(n+1)^n}{en^n}\right] $$ I've transformed it in a 0/0 inequality and tried to apply L'Hospital one time, but the ...
-2
votes
2answers
71 views

Judge the convergence of $\sum_{n=0}^\infty 1/\sqrt{n}$

How to judge the convergence of the sequence? $$\sum_{n=0}^\infty\frac{1}{\sqrt{n}}$$ Context I know two methods to judge whether a series converes: one is to calculate $\lim \frac{u_{n+1}}{u_{n}}$, ...
6
votes
4answers
162 views

Closed-forms of infinite series with factorial in the denominator

How to evaluate the closed-forms of series \begin{equation} 1)\,\, \sum_{n=0}^\infty\frac{1}{(3n)!}\qquad\left|\qquad2)\,\, \sum_{n=0}^\infty\frac{1}{(3n+1)!}\qquad\right|\qquad3)\,\, ...
3
votes
2answers
36 views

Sequence Convergence using bounding sequences

Consider the sequence $(a_n)$ with $a_n = F_{n+1}/F_n$ for $n \in \Bbb N$, where $F_n$ are the Fibonacci numbers. Show that this sequence converges to $\phi =(\sqrt{5}+1)/2$. Can someone help ...
5
votes
4answers
111 views

Limit of a sequence of products

How do you prove the following? $$\lim_{n\,\to\,\infty}\,\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots(2n)}\ =\ 0$$
2
votes
1answer
32 views

Proof of absolute convergence [duplicate]

I am independently studying calculus using MIT's publicly available materials on OCW. One of the Final Exam practice question is the following: Suppose the series $\sum_{n=1}^{\infty}a_n$ converges ...
3
votes
2answers
83 views

$\sum \tan ( 1/n)$ diverges

Show that the series $$\sum_n \tan\left(\frac{1}{n}\right)$$ diverges. I dont have any attempt to do, since I am having some troubles with series including geometric functions. I would be glad if I ...
2
votes
1answer
30 views

Convergence study of a series of functions

I am studying the convergence of the series $$ \sum_{n=0}^{\infty}\frac{\sin (x^n)}{(1+x)^n} $$ where $x \in \mathbb R$. My initial approach was to use the ratio test, but I am not getting to ...
0
votes
2answers
62 views

A identity relating a infinite series and a definite integral [duplicate]

Prove that, $$ \sum_{n=1}^{\infty} \frac{1}{n^n} = \int_{0}^{1} x^{-x}dx$$ I made no significant progress, I'm looking for hint/ideas to approach this problem. Thanks!
1
vote
2answers
40 views

sequence: use of stirling formula

I want to use the Sterling formula which says that: $lim_{n \rightarrow \infty}\dfrac{n!}{\sqrt{2*\pi}n^{n+1/2}*e^{-n}}=1$ I want to use it to show that $\lim_{n \rightarrow ...
1
vote
1answer
23 views

Test for convergence of positive term series

$$\frac{2}{1^p}\ + \frac{3}{2^p}\ + \frac{4}{3^p}\ +\ldots\,.$$ I can see that the $nth$ term is $\frac{n+1}{n^p}$ How do I test for its convergence?
7
votes
3answers
171 views

Hard Definite integral involving the Zeta function

Prove that: $$\displaystyle \int_{0}^{1}\frac{1-x}{1-x^{6}}{\ln^4{x}} \ {dx} = \frac{16{{\pi}^{5}}}{243\sqrt[]{{3}}}+\frac{605\zeta(5)}{54} $$ I was able to simplify it a bit by substituting ${y = ...
2
votes
0answers
31 views

McShane vs. Henstock-Kurzweil: Lebesgue integrable

Put in words, is it right to say that the difference of the McShane integral to the Henstock-Kurzweil integral is that the tags are not required to lie within $x_i\leq t_i\leq x_{i+1}$? If so, is ...
0
votes
1answer
24 views

Antiderivative of unbounded function?

One way to visualize an antiderivative is that the area under the derivative is added to the initial value of the antiderivative to get the final value of the antiderivative over an interval. The ...