A Proof of the Binomial Theorem, with some Applications

Extract

Many proofs of the Binomial Theorem have been given, and the proof which I give in this note has not the slightest claim to be considered new. My only object in giving it is to call attention to the fact that it depends merely on the rule of integration by parts, and provides a form of the remainder that in many applications is much simpler than any of the forms associated with Taylor's Theorem. Of its usefulness in this latter respect I give an example from the asymptotic expression for a Bessel function.