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Thank HallsofIvy. This seems like the most plausible way to handle that imo :biggrin:. Thanks.

Oh, I did expand the denominator as a power series and then integrate term by term by resulting series, but I can't proceed to the next step. The general formula for arbitrary k is too complicated for me.

Yeah, looks like it yielded some positive results finally , thanks very much for your help.

Homework Statement \int_0^1\frac{\sin(\pi x)}{1-x}dx Homework Equations \int \frac{\sin (\pi x)}{1-x}=Si(\pi-\pi x) The Attempt at a Solution I was stuck on the above integral while solving an exercise, I found out earlier on Wolfram that this integral doesn't probably have an elementary...

Have a look at this http://www.infocobuild.com/education/audio-video-courses/mathematics/math210-calculus-one-umkc.html. Also check the thread https://www.physicsforums.com/showthread.php?t=349631, which should include many more.

The field Q is not complete. R is complete. You're supposed to deal with the sup in R. The argument \sqrt{5} is not in Q should only be applied to show that the set aforementioned has no sup over Q.

Does this observation help? $\sup {x_n} = \max (\sup {y_n},\sup {z_n})$

Calculus Revisited: Complex Variables, Differential Equations, and Linear Algebra. Nice. Prof. Herbert Gross. http://ocw.mit.edu/resources/res-18-008-calculus-revisited-complex-variables-differential-equations-and-linear-algebra-fall-2011/index.htm Downloadable lecture videos and notes.

Is there a function f(x): \mathbb{R} \to \mathbb{R} such that \lim_{x \to 0} x f(x) = a \neq 0.

A rational number a/b where b is nonzero can, however, be exactly represented on the real line, can't it?. If irrational numbers didn't exist, then the the number line would have all elements being rational, which can be disproved. Then they must somehow exist :smile:.

Do you think having Bogachev's Measure Theory (vol. I) as a first exposure to measure theory sounds a good idea? I mean while I can understand well the concepts presented in the book, I find some techniques used in the proof section quite hard to follow. :confused:

Many thanks :).

I'm taking PoMA-Rudin, do I have to complete all the exercises after every chapter to be regarded as understanding the material ? Does all the tools for solving the exercises lie in the material? Because I feel many problems require more than the textbook. Thanks.

Thank you very much, I can take it from here :D.

Thanks, I didn't think thoroughly before posting this silly question, sorry.

Ah, my bad :tongue:, I meant to ask if EVERY circle having irrational radius have no lattice points on its boundary, not an example :smile:.

Thanks, but the equation x^2 + y^2 =1/2 seems to have no integer solution...

Does any circle having irrational radius have no lattice points on its boundary ? Extended question: Is there any way to determine the number of lattice points lying on the boundary of a given circle ? *The centres of these circles are all (0,0) *

Actually my original problem was determining the convergence or divergence of the following improper integral: \int^{+∞}_{0} \frac{ln(x^{2}+4^{x})}{\sqrt{3x^{7}+7x^{^3}}}dx I split the integral into \int^{+∞}_{1} \frac{ln(x^{2}+4^{x})}{\sqrt{3x^{7}+7x^{^3}}}dx and \int^{1}_{0}...

Idea: Establish a bijection f: N -> A n |-> f(n)=x_{n} If there exists no N: \forall n > N, x_{n} = const, then A must be infinite -> hence we obtain a contradiction. :smile:

Homework Statement \int \frac{ln(x^{2}+4^{x})}{\sqrt{3x^{7}+7x^{^3}}}dx Homework Equations X. The Attempt at a Solution Wolfram Alpha seem to give no answer.

Thanks, I was just confused when my answer doesn't match the exercise's solution, maybe some mistakes were made . Thank again :biggrin:.

Thanks but I'm wondering why they used the first one instead of the second. lim_{\stackrel{}{t \rightarrow 0^{+}}}\int^{1}_{t}\frac{1}{\sqrt{1-x^{2}}} lim_{\stackrel{}{t \rightarrow 1^{-}}}\int^{t}_{0}\frac{1}{\sqrt{1-x^{2}}}

We have that \int^{1}_{0}\frac{1}{\sqrt{1-x^{2}}}=lim_{\stackrel{}{t \rightarrow 0^{+}}}\int^{1}_{t}\frac{1}{\sqrt{1-x^{2}}}=lim_{\stackrel{}{t \rightarrow 0^{+}}}[arcsin(x)]^{1}_{t}=\frac{\pi}{2} However, I think \int^{1}_{0}\frac{1}{\sqrt{1-x^{2}}} should equal to lim_{\stackrel{}{t...

guess I misunderstood some of the concept in the first place, I thought the ball centered at 1 must completely lie in the interval [1/2,1]. :D. Thank guys.

Hi folks, as I was reviewing the metric space section in Amann- Escher textbook, I came across the following example of neighborhood: "For \left[0,1\right] with the metric induced from R, \left[\frac{1}{2},1\right] is a neighborhood of 1, but not of \frac{1}{2}." However I can't point out the...

Never mind, I grasped the idea at last :biggrin: , but thanks aw.

This should do it :biggrin:

Hi everyone, I'm currently having problems with these concept, my textbook states that: Let S be the set of all subsequential limits of s_n, then sup S= limsup s_n inf S= liminf s_n Knowing that S is also the set of all limits point of s_n, however I'm wondering how I could determine this...