Past exam of the options I course of the University of Oxford / 2004 / D

For $x>0$ the geometric ratio $e^{-2x}$ has modulus less than one, andEvery summand is nonnegative. The monotone convergence theorem therefore permits integration of the sum:The integral is finite because its integrand is bounded above by $e^{-x}$, whose integral is one. The displayed paired sums are the even partial sums of the alternating harmonic series; its odd partial sums differ by a term tending to zero. Thus the required equality holds with the usual series ordering. One can additionally evaluate it by $t=e^{-x}$:Grouping into nonnegative pairs supplies the justified interchange; absolute convergence of the ungrouped integrated alternating series is not needed and would be false.

The indicated measurable function is nonnegative. Tonelli's theorem applies without first assuming integrability and givesThus $h$ is integrable. Fubini's theorem now allows the reverse order. For $x>1$,ConsequentlyAt $x=1$ the quotient has the removable limit $b-a$; that endpoint does not create an additional divergence.