Re: HSC 2013 4U Marathon
i) All variables within S_k is positive, all operations are positive, hence S_k > 0
ii) Consider the upper and lower rectangle of the curve y=1/x, from x=t and x=t+b
iii) Consider the lower bound and the middle term for (ii)
Substitute, t=kn, t=kn+b, t=kn+2b, ... , t=kn+(n-1)b, add them all side by side:
Take the upper bound and middle term for (ii)
Substitute, t=kn+b, t=kn+2n, ... , t=kn+nb, and add them side by side, note that the logarithms telescope and cancel out
Then make the inequality into one whole one with b/a S_n in the middle, multiply everything by a/b to yield what is needed. Then taking the limit to infinity, makes the lower bound converge to the upper bound, by the squeeze theorem the middle term then becomes the upper and lower bound's limit.
ii) Make f(t) = e^t. Then simplify out, once integral is simplified, sub in x=1.
iii)
Integrate both sides from 0 to 1. Both are positive over this interval.
iv) (iii), proves that the error integral in (ii) converges to zero as n approaches infinity (by squeeze theorem in part (iii). Hence as n approaches infinity, for part (ii) the integral converges to zero and the sum remains.
v) general inductive proof, we do not refer to the lower bound in this proof, because due to the infinite series in iv)
by default since all terms are positive.
The upper bound is proven by induction, by using the infinite series in the assumed step, then subtracting 1/(m+1)! to both sides of the upper bound inequality.
vi) From what is proven in part (v), multiply everything by n!. The sum in the middle becomes a series of integers, hence is N.
viii) Let e=p/q, p and q are integers, since p and q are integers and n! = n(n-1)(n-2)(n-3)....(q)....(3)(2)(1). n! is divisible by q and hence p/q n! is an integer.
But since N is an integer therefore p/qn! - N is an integer, BUT the lower bound is 1/(n+1) < 1 and the upper bound is 1/n < 1. Therefore the middle CANNOT be an integer.
Hence a contradiction has arisen due to letting e=p/q
Therefore by proof by contradiction, e is irrational.