Show that if f and g are arbitrary functions of their respective arguments, then u = f(x - kt + iay) + g(x - kt - iay), is a solution of d^2u/dx^2 + d^2u/dy^2 = 1/C^2 * d^2u/dt^2, where alpha^2 = 1 - k^2/C^2.

2Medium15 marks

Find the upper and lower Riemann integrals for the function f defined on [0, 1] as follows:

f(x) = (1 - x^2)^(1/2), if x is rational.

f(x) = (1 - x), if x is irrational.

Hence, show that f is not Riemann integrable on [0, 1].

3Medium15 marks

Using Cauchy's general principle of convergence, examine the convergence of the sequence <fn>, where fn = 1 + 1/1! + 1/2! + ... + 1/n!.

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