For a measurable function $f$ on $[1,\infty)$ which is bounded on bounded sets, define $a_n = \int_n^{n+1}f$ for each natural number $n$. Is it true that $f$ is integrable over $[1,\infty)$ if and only if the series $\Sigma_{n=1}^\infty a_n$ converges? Is it true that $f$ is integrable over $[1,\infty)$ if and only if the series $\Sigma_{n=1}^\infty a_n$ converges absolutely?
I've been working on this problem for awhile now and I fear I have gotten tunnel vision... I would really appreciate some help!! This is problem #29 from Royden fourth edition on page 89. My gut is telling me the if and only if holds for absolutely convergent, otherwise there may be some voodoo possible with rearranging terms or something.
Appreciate the help!! Thanks!
Consider what happens with $f(x)=\sin(2\pi x)$.