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ThePerfectHacker · #1 September 4th, 2007, 07:04 PM

1. Let $\{a_n\}$ be a non-increasing sequence so that $\sum_{n=1}^{\infty}a_n < \infty$ . Prove that $\lim \ na_n = 0$ .

And this problem is for the younger kids. (So give them a chance to solve it).

2. Given a positive integer $n$ define a $k$ -partition to be a sum of $k$ positive integers which sum to $n$ . For example, $n=10$ . The following are $4$ -partitions. $10 = 4+4+2+2$ and $10 = 2+2+4+4$ and $10 = 1+1+1+7$ . Notice that $2+2+4+4\mbox{ and }4+4+2+2$ are considered distinct*. Say you a given a specific $n$ . And given a specific value of $k$ , can you find the total number of $k$ -partitions of this integer, with a formula?** Now try to see how many partitions (again not counting order) exist for a given integer $n$ (the answer is really supprising).
Hint: Review your Combinatorics formula for this one.

*)This is done to simplify this problem. When these are not considered distinct this forms a problem from number theory called the "partition problem". It is a very complicated problem.

**)The problem is not whether you can. But rather how you can. If it was the later the answer would be "yes" turning this problem into a worthless problem.