Question

Suppose g(x) is continuous on [a,b]. Justify that ∫[a,∞] g(x)dx converges exactly if ∫[b,∞] g(x)dx does.

Answer 1

Pretty easy to do..

\[\int_a^\infty g(x)dx = \int_a^bg(x)dx + \int_b^\infty g(x)dx\]

A definite proper integral will always have a specific value if the function is continuous and differentiable on that interval, so \(\int_a^bg(x)dx\) will just be some constant and the convergence will therefore depend on the \(\int_b^\infty g(x)dx\) part.