Math problems

Q1
We need the help of Bertrand's postulate.
Bertrand's postulate (actually a theorem) states that for each n ≥ 2 there is a prime p such that n < p < 2n.

Now for each n, if n is even , there is a prime number n/2 < p < n,
If n is odd, there is a prime number (n+1)/2 < p < n+1.
In each case, 2p > n, p is not a factor of any integer k from 1 to n, except k = p. But as p <= n, p is a factor of n!.
Therefore n! / p is an integer.
Let s = 1/2+1/3+...+1/n,
s x (n! / p) = Σ(k=0,…,n ; k =/= p) n! / p / k + n! / p^2
for k not equal to p, n! / p / k is an integer. However, n!/P^2 is not an integer as p is only a factor of p but not other integers from 1 to n.
There s is not an integer.

Q2
Reference : http://en.wikipedia.org/wiki/N-sphere
The recurrence relation for Vn(R) can be proved via integration with 2-dimensional polar coordinates:For small values of n, the volumes, Vn, of the n-ball of radius R are:
圖片參考：http://upload.wikimedia.org/math/7/8/2/7827ccb8781c203b72f5ea435cc56bad.png




2010-05-18 15:00:25 補充：
Q3. I can't help.

Pure Maths 2010 Paper: http://lsforum.net/board/thread-134582-1-3.html