I wonder why would a C++, C#, Java developer want to learn a dynamic language?
Assuming the company won't switch its main development language from C++/C#/Java to a dynamic one what use is there for a dynamic language?
What helper tasks can be done by the dynamic languages faster or better after only a few days of learning than with the static language that you have been using for several years?
Update
After seeing the first few responses it is clear that there are two issues.
My main interest would be something that is justifiable to the employer as an expense.
That is, I am looking for justifications for the employer to finance the learning of a dynamic language. Aside from the obvious that the employee will have broader view, the
employers are usually looking for some "real" benefit.
Towards answering the updated question, its a chicken/egg problem.  The best way to justify an expense is to show how it reduces a cost somewhere else, so you may need to spend some extra/personal time to learn something first to build some kind of functional prototype.
Show your boss a demo like "hey, i did this thing, and it saves me this much time [or better yet, this much $$], imagine if everyone could use this how much money we would save"
and then after they agree, explain how it is some other technology and that it is worth the expense to get more training, and training for others on how to do it better.
Why learn Perl, Python, Ruby if the company is using C++, C# or Java as the application language?