IANAL, but wouldn't that be against the rule that character evidence cannot be used by the prosecution (unless in countering character evidence from the defence)?
Specifically, I believe what you describe would be in contravention of the following: https://www.law.cornell.edu/rules/fre/rule_404
> (a) Character Evidence.
> (1) Prohibited Uses. Evidence of a person’s character or character trait is not admissible to prove that on a particular occasion the person acted in accordance with the character or trait.
> [...]
> b) Other Crimes, Wrongs, or Acts.
> (1) Prohibited Uses. Evidence of any other crime, wrong, or act is not admissible to prove a person’s character in order to show that on a particular occasion the person acted in accordance with the character.
> (2) Permitted Uses. This evidence may be admissible for another purpose, such as proving motive, opportunity, intent, preparation, plan, knowledge, identity, absence of mistake, or lack of accident.
You can't testify that a defendent was "a jealous man" or "of unsavory character". You can testify that you saw him shake an infant angrily until she passed out.
In other words, you can't argue that given there is evidence that the defendant shook babies in anger before, he is the kind of guy who shakes babies in anger, and therefore this is evidence in favour of him having shaken this baby to death.
Regarding (b)(2), I interpret that the testimony that the defendant shook infants in anger in the past would only be admissible under (b)(2) if the previous acts were "proving motive, opportunity, intent, preparation, plan, knowledge, identity, absence of mistake, or lack of accident". I cannot see how having shaken an infant in the past could provide evidence for any of the above as relates to the separate and specific alleged infant-shaking event for which the defendant was on trial.