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Limit of an Infinite Series [7/18/1995]
What is the limit of (n!)^(1/n)?

Lines, Points, and Infinities [09/01/2001]
What is the cardinality of the set of real numbers between 0 and 1? Is this cardinality less than, greater than, or equal to the cardinality of real numbers between 0 and 2?

Pi to x Million Decimal Places [8/1/1995]
How is pi calculated to x million decimal places?

Predictor-Corrector Methods [09/07/2002]
Could you please explain what a predictor-corrector method is, possibly using Euler's and Huen's methods as examples?

Rational and Irrational Numbers [11/12/1997]
Which set is bigger, the set of rational or irrational numbers?

Rational Numbers [11/24/1997]
Which is greater, the number of rational numbers between 0 and 1 or the number of rational numbers between 0 and 2?

0 Raised to a Power [04/25/2002]
What is the limit of 0^n as n approaches infinity? Can you explain why it equals zero or why it can't equal zero, or if it's undefined?

1 + 2 + 3 + 4 + ... Equals ... -1/12?! [09/18/2012]
Doctor Vogler explains how functions extended by analytic continuation can be evaluated to produce counterintuitive results.

Analysis [11/29/1994]
In analysis: If f:[0,1] is continuous. Show that there is an x in [0,1] such that f(x) = x. Problem #2: If A and B are open and closed sets respectively, of R^n, show B\A is closed and A\B is open.

Analysis and the Derivative [01/23/2001]
Suppose that f:RtoR is differentiable at c and that f(c)=0. Show that g(x):=|f(x)| is differentiable at c if and only if f'(c)=0.

Analysis Questions [12/07/1996]
I have been going over some of my professor's old exams and can't figure out some of the questions.

Banach-Tarski Paradox [07/20/2002]
The Banach-Tarski paradox decomposes a sphere of a fixed radius into a finite number of parts and then puts them together again to form two spheres, each with the same radius as the original sphere. How is this possible?

Best-fitting Line to a Number of Points [04/25/2001]
I have a number of points on a plane and want to find a line that best- fits through the points, minimizing the sum of squares of the distances of each point from the line.

Cardinality, Area, and Probability [09/05/2003]
Divide a rectangle into two regions, one with twice the area of the other. What is the probability that a point randomly selected will fall in the larger area? Is it 2/3, because the larger region contains 2/3 of the area? Or is it 1/2, because the two regions contain the same number of points?

Cauchy Sequences [09/11/1997]
I'm trying to prove that every Cauchy sequence has a sequential limit point.

Chaos Theory and Converging Sequences [2/14/1996]
I'm investigating a process of dividing and adding - take a number, divide by 5, add 2, write down the result, repeat the process using the result - and realise that using this process my sequence will converge towards 2.5, where nothing else can be done to it. By reversing the process - adding 2 first then dividing by 5 - my sequence will diverge to what I guess will be infinity. What else can I say?

Chaotic Functions [10/30/2000]
Can you give some mathematical examples of chaos theory?

Closure and Compactness in a Metric Space [10/08/2002]
Regard Q, the set of all rational numbers, as a metric space, with d(p,q)=|p-q|... Show that E is closed and bounded in Q, but that E is not compact. Is E open in Q?

Compact Sets [02/04/2001]
Let A be an infinite set in the real numbers with a single accumulation point in A. Must A be compact?

Compact Sets and Hausdorff Spaces [03/19/2003]
How do you prove that every compact subset of a metric space is closed?

Complex Analysis Geometry Proof [09/28/2004]
I'm looking for a proof that arg z1 + arg z2 = arg(z1z2).

Complex Integrals and the Residue Theorem [12/18/2000]
How can I calculate the integral over C of (z^2/((z-1)^2*(z+1)))dz, where C is the circle C = {z| |z - 2i| = 2}? Can I use the Taylor Series?

Continuity [11/24/2001]
Suppose f and g are continuous on I = [a,b]... Prove there exists a k greater than 0 such that f(x) + k is less than or equal to g(x) for all x in I.

Continuous but Not Differentiable? [12/27/2003]
I have heard that Weierstrauss found a function that is continuous everywhere, but differentiable nowhere. How is that possible?

Continuous Function [05/18/1999]
Is there a way of constructing a continuous function that is nowhere differentiable?

Counting Infinite Sets [06/02/1997]
What is the cardinality of all the pieces that are part of the real number line?

Definition of the Limit [11/05/2002]
I need some help with the definition of the limit, particularly choosing delta for a given epsilon.

Dense and Nowhere Dense Sets [04/25/1999]
Can you define the mathematical terms 'dense' and 'nowhere dense'?

Deriving the Gamma Function [12/15/2000]
How can you prove that sqrt(pi)/2 = (1/2)!, and what is a fractional factorial like that equal to?

Distance between Lines [03/04/1997]
Given the equations of several lines in a room, determine the distance between them.

Elliptical Arc Length [11/17/1999]
Is there a series approximation or some other numerical method for calculating the arc length of an ellipse?

Epsilon and Delta [04/18/1999]
How can I show that sin(x^2) is not uniformly continuous on the reals?

Euler Formula: e^(pi*i) = -1 [6/5/1996]
Why does e^(pi*i) = -1?

Explain Supremum [02/02/1998]
Can you please explain, perhaps with an example, the concept of "supremum"?

Fibonacci sequence [1/28/1996]
What is the explicit formula for the Fibonacci numbers?

Finding Repetitive Patterns in Economic Data [1/28/1995]
What branch/subset of mathematics do I need to learn so that I can write computer software that will identify and/or search pairs of data for specific patterns?

Fixed-Point Theorem [05/02/2002]
An intuitive reason why the theorem is true, and a proof.

Formal Definition of a Limit [08/01/1998]
What is the connection between the open intervals and the informal definition of the limit?

Formulas for N-Dimensional Spheres [10/26/2000]
What is the next term in the sequence pi*r^2, (4/3)pi*r^3, ...? Would it be the formula for the volume of a 4-dimensional sphere?

Fourier Series and the Zeta Function [08/13/1998]
How do you evaluate Zeta of 2?

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