• ### Integration sans fundamental theorem of calculus?

Hope you all had a Merry Christmas yesterday. The challenge du jour: Let n be an arbitrary fixed positive integer, and consider the right-hand Riemann sums of the function x^n obtained from a partition of [0, 1] into m equal intervals. These are: [k=1, m]∑(1/m * (k/m)^n) Using the fundamental theorem of calculus, it is of course trivial to... show more
Hope you all had a Merry Christmas yesterday. The challenge du jour: Let n be an arbitrary fixed positive integer, and consider the right-hand Riemann sums of the function x^n obtained from a partition of [0, 1] into m equal intervals. These are: [k=1, m]∑(1/m * (k/m)^n) Using the fundamental theorem of calculus, it is of course trivial to prove that: [m→∞]lim [k=1, m]∑(1/m * (k/m)^n) = [0, 1]∫x^n dx = x^(n+1)/(n+1) |[0, 1] = 1/(n+1). Your mission, should you decide to accept it, is to prove the limit [m→∞]lim [k=1, m]∑(1/m * (k/m)^n) = 1/(n+1) _directly_, without using the fundamental theorem. Can you do it?
• ### I have a slightly harder mathematical puzzle for you...?

Find constants a, b, c, and d such that: 4x³ - 3x + √3/2=a * (x-b) * (x-c) * (x-d) Rules: #1: Your factors must be exact. No numerical approximations allowed. #2: Neither the imaginary constant, nor the square root of any negative number may appear in your answer. #3: Your answer must be simple. Nothing more complicated than a trigonometric... show more
Find constants a, b, c, and d such that: 4x³ - 3x + √3/2=a * (x-b) * (x-c) * (x-d) Rules: #1: Your factors must be exact. No numerical approximations allowed. #2: Neither the imaginary constant, nor the square root of any negative number may appear in your answer. #3: Your answer must be simple. Nothing more complicated than a trigonometric function may appear in your answer. #4: Yes, this problem is solvable.
• ### A mathematical experiment?

f(x) is a monic polynomial of degree 6. It has the following values: f(0)=0 f(1)=1 f(2)=2 f(3)=3 f(4)=4 f(5)=5 Question 1: what is f(6)? Question 2: do you think that other people are likely to get this question wrong?
f(x) is a monic polynomial of degree 6. It has the following values: f(0)=0 f(1)=1 f(2)=2 f(3)=3 f(4)=4 f(5)=5 Question 1: what is f(6)? Question 2: do you think that other people are likely to get this question wrong?
• ### What the heck does "pls" mean?

I see that word a lot after many of the questions in this section, and I'm wondering what it means. I know it doesn't mean please, because surely anyone who is going to ask people to take valuable time out of their day to do their homework for them would not be so ill-mannered as to mock all standards of politeness and decency by... show more
I see that word a lot after many of the questions in this section, and I'm wondering what it means. I know it doesn't mean please, because surely anyone who is going to ask people to take valuable time out of their day to do their homework for them would not be so ill-mannered as to mock all standards of politeness and decency by deliberately misspelling the word "please," so I assume that it must be some kind of internet abbreviation with which I am not familiar. So what does it mean?
• ### Would Flatland be more widely remembered...?

Some of you may be familiar with the old classic "Flatland: A Romance of Many Dimensions" by Edwin Abbott (if you aren't, you can read it at http://xahlee.org/flatland/index.html ). It's really quite good, and an excellent introduction to higher-dimensional thinking, but a lot of people have never heard of it. So what I want to... show more
Some of you may be familiar with the old classic "Flatland: A Romance of Many Dimensions" by Edwin Abbott (if you aren't, you can read it at http://xahlee.org/flatland/index.html ). It's really quite good, and an excellent introduction to higher-dimensional thinking, but a lot of people have never heard of it. So what I want to know is: do you think Flatland would be more widely remembered if it had been called "Shapes on a plane"?