18.014 Final Comments and Grades

Comments on quiz 4. The last quiz didn't go as well as the previous three; the median grade was 64. Overall I was disappointed in the performance on quiz 4 but quite happy with how the course as a whole went.
Answer to last question.Here's why 1^100 / 2 + 2^100/ 2^2 + 3^100 / 2^3 + ... is rational.

Lemma: You can check that if p(x) is a polynomial with rational coefficients, then

x d/dx ( p(x) / (1-x)^n ) = q(x) / (1-x)^(n+1),
where q(x) is also a polynomial with rational coefficients.

Note that 1+x+x^2 + ... = 1/(1-x); this converges for x=1/2 (or indeed for any x between -1 and 1). Show that if you apply x d/dx to 1/(1-x) 100 times, and plug in x=1/2, you get the desired sum, and (using the lemma) that this is rational.

(That is the kernel of the solution; if you'd like to hear more details, just ask!)

(Congratulations to Russell Moriarty for solving this very hard problem!)

Final grades

Here are the final grades to those people who left alphanumeric codes (as far as I could tell). If you're not on the list and would like your grade, please drop by, or e-mail. (Keep in mind that e-mail is not fully secure.) Also, if you would like to know more details of your grade, please ask as well.

Code Quiz 4 grade Overall grade
1024 96 A+
3837701826 57 C+
4410849 58 B+
5NGK 82 A
9666703 78 A
A27L 55 C+
A985586 58 A-
ABC123 84 A
Blue32 59 A-
D007 71 A-
FU42 38 C-
gr8score 35 C-
GS57 61 A-
JAC 79 A
joondol 65 A-
kermit 28 C-
oltugaykut 94 A+
rax65537 64 C-
S21 85 A+
SPX0873 94 A
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