extra credit
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3/3
Here are five interesting problems on recursions. If you can solve them, you will earn 10 points and achieve a higher level of skills. Note that the due day for all bonus problems is March 16. -
2/13
In the discussion of expected values of flipping coins, we have derived the following inequality :
For 0< p <1, and a positive integer n,
∑k=1n k ( n
k) pk(1-p)n-k = np
(The proof should be complete and rigorous. ) -
1/24
Try to draw Venn diagram of five sets. -
01/10/06
In the discussion of the series, the sum from 1 to infinity 1/(k(k+1)), i.e.,
∑k=1∞ 1
− − −
k(k+1)
Can you pack rectangles of size 1/k by 1/(k+1) into a unit square?
(Note that "pack" means not overlapping and "unit square" has sides of length 1.)
A dinner for a complete answer.
$5 for a successful packing of such rectangles of sizes up to k=10,000.
No due day. The first person who has an answer gets the award.