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# Prove or disprove: If f is positive and lim In = A, then ∫ f(x) converges to A

The following function is defined for all , and is a positive integer. Prove or provide a counterexample to the following statement.

If is positive and if then Incomplete.

1. Evangelos says:

After looking at the counterexample for exercise 25, I’ll have to rework this one because is not a necessary condition for convergence. There’s actually a pretty simple proof for this one since is set to be positive for all Proof. If is positive for all , then, for any for some integer , the integral is bounded above by . Thus, if as , then so does .

2. Evangelos says:

Proof. Since takes positive values for all , for to converge as , must go to zero as . But if as and , then, following the proof written in exercise 21, converges and has value A.

• Evangelos says:

I think I broke the Latex interpreter… lol. That mess of script is supposed to read: 