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.**

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Stumbling Robot

A Fraction of a Dot
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Prove or disprove: If *f* is positive and *lim I*_{n} = A, then *∫ f(x)* converges to *A*

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### Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):

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.**

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 .

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.

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