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Browse High School Analysis

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Proofs of e [03/21/2002]
I found out that the definition of e is: e = lim, as n approaches infinity (1+1/n)^n. Is there a proof for this or for 1+2(e-1/n)?

Prove S' Closed in R^n [10/07/2002]
Prove that S' (the derived set or set of all accumulation points of S) is closed in R^n.

Prove That a Set Is Uncountably Infinite [10/31/1997]
Prove that the set [0,1]x[0,1] is uncountably infinite.

Prove the Commutative Law [4/1/1996]
I have a question that may sound very silly: can anyone PROVE the commutative law?

Proving a Set is Closed [04/15/1998]
Let f be a continuous real valued function on Rn, and let S = {x in Rn : f(x) = 1}. Prove that S is a closed subset of Rn.

Proving/Deriving the Quadratic Equation [4/1/1996]
How can one derive the quadratic equation, i.e. prove it?

Proving Limits at Infinity Using the Formal Definition [04/11/2004]
We are required to find a limit (informally), then prove that our answer is correct by using the formal definition of limits. I'm not sure I'm on the right track.

Proving Limits Formally [01/04/1999]
Given a sequence, I can guess what the limit it. How do I actually prove my guess is the limit?

Proving the Properties of Natural Numbers [03/08/2000]
How can you prove or derive the commutative, associative, and distributive properties of numbers?

Rational/Irrational Numbers [03/12/1999]
Prove that between any two numbers there are a rational and an irrational number and, therefore, that there are infinite numbers of each.

Real Analysis [9/13/1996]
Let I be a subset of R where I is an interval and let f map I to R be increasing on I...

Real Plane, Complex Plane [09/16/2002]
When does Az + Bz + c = 0 become a straight line?

Set Notation via Broken Typewriter [11/19/2002]
I have a machine that only types out ones (no spaces or tricks involved). What procedure must you do to this machine to get any given finite set? For example [2,85,11,5,60]. For a different set, the number of ones that the machine types out will vary.

Significant Difference? [05/08/1999]
A research question: does mean delivery time differ for boy babies and girl babies?

Singularity [03/01/1999]
What is a singularity?

Sizes of Infinities [01/31/1997]
How can you prove that one infinity is larger than another?

Solving the Equation x^y = y^x [12/09/2004]
Solve x^y = y^x for x in terms of y only. Also, how do I find all possible solutions beyond the obvious ones of x = y, (2,4), and (4,2)?

A Sorting Lower Bound [10/01/1998]
Show that you cannot sort 3 elements with 2 comparisons, an example of lower bound.

Sum of Convergent Series [09/25/1999]
How can you find the sum for k = 0 to infinity of 1/[(k+1)(k+3)], and the sum for k = 0 to infinity of [(25/10^k) - (6/100^k)]?

Teaching Synthetic Division [10/23/1996]
Do you have any advice for teaching synthetic division, on the computer, to the rest of my class?

Two Definitions of Limits, with Examples [05/11/1998]
Epsilon-delta definitions of the limit of a function and the limit of a sequence.

An Uncountable Set [09/28/1998]
How do you prove, using diagonalization, that the set of functions from N to N is uncountable?

Using L'Hopital's Rule [01/22/1999]
Prove that e^(-1/3) is the limit for y =(arctan(x)/x)^(1/(x^2)) when x -> 0.

Using Newton's Method to Solve an Implicit Equation [10/14/2007]
How can I solve the equation (sin(X))/X = 0.7031? My book gives the answer but does not explain how to find it.

Weierstrass Curve [09/13/2002]
About 100 years ago the mathematician Weierstrass gave an example of a curve consisting of angles, or corners, and nothing else. Where can I find this equation?

What is 0.999... + 0.999...? [09/27/1999]
How can I show that 0.999... + 0.999... = 2?

What is 1^infinity? [12/10/1998]
Why is 1^infinity indeterminate? How can you show this using limits? Why is it not equal to 1, as intuition leads us to believe?

X^Y = Y^X [11/13/2002]
Find (with proof) all pairs of integers {x,y} that satisfy x^y=y^x.

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