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’?
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.
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.
It is hard to know what there is to prove here.
Note that because you know that 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.)