Search found 77 matches

by Demetres
Tue Jul 12, 2016 7:22 am
Forum: Complex Analysis
Topic: Contour Integration
Replies: 1
Views: 2535

Re: Contour Integration

Let \[f(z) = \frac{\sin(\pi z^2) + \cos(\pi z^2)}{z^2 - 3z+2}.\] Then \(f\) has exactly two simple poles inside the countour of integration, at \(z=1\) and at \(z=2\). So the integral \(I\) is equal to \(2\pi i (\mathrm{Res}(f;z=1) + \mathrm{Res}(f;z=2)).\) We have \[ \mathrm{Res}(f;z=1) = \lim_{z\t...
by Demetres
Tue Jul 12, 2016 7:05 am
Forum: Real Analysis
Topic: Largest prime factor
Replies: 2
Views: 2325

Re: Largest prime factor

Note that \[ \frac{2012}{k(k+1)\cdots (k+2012)} = \frac{1}{k(k+1)\cdots(k+2011)} - \frac{1}{(k+1)(k+2)\cdots (k+2012)}.\] So the sum \[ \sum_{n=1}^{\infty} \frac{2012}{n(n+1)\cdots (n+2012)}\] is telescopic and equals \(1/(2012)!\). In particular the expression of the question is equal to \(2012!\) ...
by Demetres
Sat Jul 09, 2016 9:01 am
Forum: Real Analysis
Topic: Maximum value of Ratio
Replies: 1
Views: 1760

Re: Maximum value of Ratio

By Hölder's inequality, and since \(f\) is positive, we have \[ \int f(x) \, dx \leqslant \left( \int f(x)^3 \, dx\right)^{1/3} \left( \int 1^{3/2} \, dx\right)^{2/3}.\] It immediately follows that \(R \leqslant 1\) and we can have equality when \(f\) is (for example) constant. [In fact, since we ar...
by Demetres
Sat Jul 09, 2016 5:37 am
Forum: Real Analysis
Topic: A known exercise
Replies: 1
Views: 1599

Re: A known exercise

Making the substitution \(y = x^3\) we obtain \[ I = \frac{1}{3} \int_0^1 y^{-2/3}(1-y)^{-1/2} \, dy.\] It is well known that \[ B(m,n) = \int_0^1 x^{m-1}(1-x)^{n-1} \, dx = \frac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)}.\] So \(I = \frac{\Gamma(1/3)\Gamma(1/2)}{3\Gamma(5/6)}.\) A theorem of Chebyshev says ...
by Demetres
Thu Jul 07, 2016 1:51 pm
Forum: Calculus
Topic: \(\sum_{n\geq1}\frac{1}{(2n-1)(3n-1)(4n-1)}\)
Replies: 3
Views: 3258

Re: \(\sum_{n\geq1}\frac{1}{(2n-1)(3n-1)(4n-1)}\)

We begin by observing that \[ \frac{1}{(2n-1)(3n-1)(4n-1)} = \frac{2}{2n-1} - \frac{9}{3n-1} + \frac{8}{4n-1}.\] We now define \[ f(x) = \sum_{n=1}^{\infty} \left( \frac{2x^{12n-6}}{2n-1} - \frac{9x^{12n-4}}{3n-1} + \frac{8x^{12n-3}}{4n-1} \right).\] This is a power series which converges absolutely...
by Demetres
Thu Jul 07, 2016 1:42 pm
Forum: Analysis
Topic: A class of alternate infinite series
Replies: 2
Views: 3115

Re: A class of alternate infinite series

Let us begin by recalling that the Beta function satisfies \[ B(x,y) = \int_0^1 t^{x-1}(1-t)^{y-1} \, dt = \frac{(x-1)!(y-1)!}{(x+y-1)!}\] whenever \(x,y\) are positive integers. In our case we have \[ \sum_{n \geqslant 1} \frac{(-1)^{n-1}}{n(n+1) \cdots (n+k)} = \frac{1}{k!} \sum_{n\geqslant 1} (-1...
by Demetres
Thu Jul 07, 2016 1:18 pm
Forum: Real Analysis
Topic: Exponential Product in the Unit Cube
Replies: 3
Views: 2714

Re: Exponential Product in the Unit Cube

George, I have also thought about this. Ypur inequalities are correct. I only include an outline of the proof: Let \[U_m = \left( \frac{1}{m^{n-1}},m\right)^n \subseteq \mathbb{R}^n.\] Suppose we want to maximize \((1+x_1)^{1/x_1} \cdots (1+x_n)^{1/x_n}\) on \(\overline{U_m}\) subject to the conditi...
by Demetres
Thu Jul 07, 2016 1:16 pm
Forum: Real Analysis
Topic: Exponential Product in the Unit Cube
Replies: 3
Views: 2714

Re: Exponential Product in the Unit Cube

The claim is false for every \(n \geqslant 4.\) Take \(x_1 = \cdots = x_{n-1} = \tfrac{1}{m}\) and \(x_n = m^{n-1}\) for some \(m\). Then \[ (1+x_1)^{\frac{1}{x_1}} \cdots (1+x_n)^{\frac{1}{x_n}} = \left(1 + \frac{1}{m} \right)^{(n-1)m}(1+m^{n-1})^{\frac{1}{m^{n-1}}}\] which tends to \(e^{n-1}\) as ...
by Demetres
Thu Jul 07, 2016 12:29 pm
Forum: Calculus
Topic: Derivative of a Power Series
Replies: 2
Views: 2746

Re: Derivative of a Power Series

Just to add that the evaluation of the derivative at \(\pi/2\) is allowed since within the circle of convergence we can determine the derivative by differentiating term by term.
by Demetres
Thu Jul 07, 2016 12:19 pm
Forum: Analysis
Topic: Non periodic function!
Replies: 3
Views: 3779

Re: Non periodic function!

I claim that \( \sin(p(x))\) is periodic if and only if \(p\) is linear (or constant). The "if" part is obvious. For the "only if" part we work similarly as my answer above. If it was periodic then its derivative \(p'(x)\cos(p(x))\) would also be periodic and since it is continuo...