Need some help on this.

Improper Reimann Integration

Suppose that f: [1,) R is locally integrable
on [1,) and f(x)0 for all x in [1,). Prove that
∫(1 to ∞)f(x)dx converges iff there existsM > 0 s.t.
∫(from 1 to x) f(t)dt M for all x in [1,).

f is said to be locally integrable on (a,b) iff f is integrable on each closed subinterval [c,d] of (a,b).

Thanks.

2. Please review your post. I think that there may be a typo in the definition of the function. Also, what is the definition of ‘locally integrable’?

3. Need some help on this.

Improper Reimann Integration

Suppose that f: [1,) R is locally integrable
on [1,) and f(x)0 for all x in [1,). Prove that
∫(1 to ∞)f(x)dx converges iff there existsM > 0 s.t.
∫(from 1 to x) f(t)dt M for all x in [1,).

f is said to be locally integrable on (a,b) iff f is integrable on each closed subinterval [c,d] of (a,b).

Thanks.

4. It is hard to know what there is to prove here.
Note that because $f(t) > 0$ you know that $F(x) = \int\limits_1^x {f(t)dt}$ is an increasing function. Knowing that the integral converges let M be the limit. Therefore, M is the bound. If we know that the integral is bounded by some M then use sequences to show that the integral converges. (Bounded increasing sequences converge.)