This is a rather nice question that involves proving the irrationality of e and pi without the usual proof by contradiction.
(a) Show that y=1+x/[(1!*a)]+(x^2)/[a(a+1)*2!]+... satisfies the differential equation xy''+ay'=y
(b) Hence or otherwise, show that y/y' is irrational
(c) Show that y/y'=(x^0.5)/tanh(2*x^0.5), and hence show that e^r is irrational, where r is a rational number
(d) By letting x=(-q^2)/4, where q is rational and non zero, show that pi is irrational
(a) Show that y=1+x/[(1!*a)]+(x^2)/[a(a+1)*2!]+... satisfies the differential equation xy''+ay'=y
(b) Hence or otherwise, show that y/y' is irrational
(c) Show that y/y'=(x^0.5)/tanh(2*x^0.5), and hence show that e^r is irrational, where r is a rational number
(d) By letting x=(-q^2)/4, where q is rational and non zero, show that pi is irrational
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