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

If is monotonically decreasing and if

exists, then the improper integral

converges.

**Incomplete.**

Skip to content
#
Stumbling Robot

A Fraction of a Dot
#
Prove or disprove: If *f* is monotonic decreasing and *lim I*_{n} exists then *∫ f(x)* converges

###
One comment

### 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 monotonically decreasing and if

exists, then the improper integral

converges.

**Incomplete.**

I think this proof holds… Would be happy to see any counterexamples.

Proof. If f is monotonic decreasing and the limit of exists as n goes to infinity, then f must go to zero as n goes to infinity. And since f is monotonic decreasing for all x, for any ,

But since f(n) goes to zero as n goes to infinity, is thus bounded by for all as n goes to infinity. Thus, is convergent for all x as x goes to infinity.